On α-adjacency energy of graphs and Zagreb index

Abstract

Let A(G) be the adjacency matrix and D(G) be the diagonal matrix of the vertex degrees of a simple connected graph G. Nikiforov defined the matrix $A_{\alpha }(G)$ of the convex combinations of D(G) and A(G) as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),$ for $0\le \alpha \le 1.$ If ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n$ are the eigenvalues of $A_{\alpha }(G)$ (which we call α-adjacency eigenvalues of G), the α-adjacency energy of G is defined as $E^{A_{\alpha }}(G)={\sum }_{i=1}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|,$ where n is the order and m is the size of G. We obtain upper and lower bounds for $E^{A_{\alpha }}(G)$ in terms of the order n, the size m and the Zagreb index Zg(G) associated to the structure of G. Further, we characterize the extremal graphs attaining these bounds.

Keywords:

• Adjacency matrix
• Laplacian (signless Laplacian) matrix
• degree regular graph
• α-adjacency matrix
• α-adjacency energy

AMS SUBJECT CLASSIFICATION:

• 05C50
• 05C12
• 15A18

1. Introduction

A simple graph is denoted by $G(V(G),E(G)),$ where $V(G)=\{v_1,v_2,\cdots ,v_n\}$ is its vertex set and $E(G)$ is its edge set. The order and size of G are $|V(G)|=n$ and $|E(G)|=m$ respectively. The set of vertices adjacent to $v\in V(G),$ denoted by N(v), refers to the neighborhood of v. The degree of v, denoted by $d_G(v)$ (we simply write d_v if it is clear from the context) is the cardinality of N(v). A graph is regular or degree regular if all of its vertices are of the same degree. The adjacency matrix $A(G)=(a_{ij})$ of G is a (0, 1)-square matrix of order n whose (i, j)-entry is equal to 1, if v_i is adjacent to v_j and equal to 0, otherwise. If $D(G)=\mathit{diag}(d_1,d_2,\cdots ,d_n)$ is the diagonal matrix of vertex degrees, the matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are the Laplacian and the signless Laplacian matrices of the graph G, respectively. Spectrum of L(G) is the Laplacian spectrum and the spectrum of Q(G) is the signless Laplacian spectrum. The matrices L(G) and Q(G) are real symmetric and positive semi-definite. For G, we take $0={\mu }_n\le {\mu }_{n-1}\le \cdots \le {\mu }_1$ and $0\le q_n\le q_{n-1}\le \cdots \le q_1$ to be the Laplacian and the signless Laplacian eigenvalues, respectively. For other standard notations, we refer to [1–4].

Nikiforov [5] introduced the concept of merging A and Q spectral theories by taking $A_{\alpha }(G)$ as the convex combinations of D(G) and A(G), and defined $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),$ for $0\le \alpha \le 1.$ Since $A_0(G)=A(G),2A_{\frac{1}{2}}(G)=Q(G),A_1(G)=D(G)$ and $A_{\alpha }(G)-A_{\beta }(G)=(\alpha -\beta )L(G),$ any result regarding the spectral properties of $A_{\alpha }$ matrix, has its counterpart for each of these particular graph matrices. Since the matrix $A_{\alpha }(G)$ is real symmetric, all its eigenvalues are real and can be arranged as ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n.$ The largest eigenvalue ${\rho }_1$ (or simply $\rho (G)$) is called the spectral radius. As $A_{\alpha }(G)$ is nonnegative and irreducible, by the Perron-Frobenius theorem, $\rho (G)$ is unique(for $\alpha \ne 1$) and there is a unique positive unit eigenvector X corresponding to $\rho (G),$ which is called the Perron vector of $A_{\alpha }(G).$ Further results on spectral properties of the matrix $A_{\alpha }(G)$ can be found in [5–15].

Gutman [16] defined the energy of a graph G as $E(G)={\sum }_{i=1}^n\left|{\lambda }_i\right|,$ where ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$ are the adjacency eigenvalues of G. Gutman et al. [17] defined the Laplacian energy of a graph G as $LE(G)={\sum }_{i=1}^n\left|{\mu }_i-\frac{2m}{n}\right|,{\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_n$ are the Laplacian eigenvalues of G. For more details, see [18]. Likewise, Abreu et al. [19] defined the signless Laplacian energy of a graph G as $QE(G)={\sum }_{i=1}^n\left|q_i-\frac{2m}{n}\right|,$ where $q_1\ge q_2\ge \cdots \ge q_n$ are the signless Laplacian eigenvalues of G and $\frac{2m}{n}$ is the average degree of G. For recent works on the energy, the Laplacian energy and the signless Laplacian energy, we refer to [20–23].

Let $s_i={\rho }_i-\frac{2\alpha m}{n}$ be the auxiliary eigenvalues corresponding to the eigenvalues of $A_{\alpha }(G).$ The α-adjacency energy $E^{A_{\alpha }}(G)$ [24] of a graph G is defined as the mean deviation of the values of the eigenvalues of $A_{\alpha }(G),$ that is, (1.1) $$E^{A_{\alpha }}(G)=\sum _{i=1}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|=\sum _{i=1}^n\left|s_i\right|.$$(1.1) It is easy to see that ${\sum }_{i=1}^n{s}_i=0.$ From the definition, it is clear that $E^{A_0}(G)=E(G)$ and $2E^{A_{\frac{1}{2}}}(G)=QE(G).$ Therefore, it follows that α-adjacency energy of a graph G merges the theories of (adjacency) energy and signless Laplacian energy. As such it will be interesting to study the quantity $E^{A_{\alpha }}(G).$

The rest of the paper is organized as follows. In Sec. 2, we obtain the upper bounds for $E^{A_{\alpha }}(G)$ and characterize the extremal graphs attaining these bounds. In Sec. 3, we obtain the lower bounds for $E^{A_{\alpha }}(G)$ and characterize the extremal graphs attaining these bounds.

2. Upper bounds for α-adjacency energy of a graph

Let ${\mathbb{M}}_{m\times n}(\mathbb{R})$ be the set of all m × n matrices with real entries, that is, ${\mathbb{M}}_{m\times n}(\mathbb{R})=\{X:X={(x_{ij})}_{m\times n},x_{ij}\in \mathbb{R}\}.$ For $M\in {\mathbb{M}}_{m\times n}(\mathbb{R}),$ the Frobenius norm is defined as $$\parallel M{\parallel }_F=\sqrt{\sum _{i=1}^n\sum _{j=1}^n|m_{ij}{|}^2}=\sqrt{\mathit{trace}(M^{t}M)},$$ where trace of a square matrix is defined as sum of the diagonal entries. Further, if $M{M}^t=M^{t}M,$ then $\parallel M{\parallel }_F^2={\sum }_{i=1}^n|{\lambda }_i(M){|}^2,$ where λ_i is the i^th eigenvalue of the matrix M.

The Zagreb index Zg(G) of a graph G is defined as the sum of the squares of vertex degrees, that is, $Zg(G)=\sum _{u\in V(G)}{d}_G^2(u).$

The following lemma can be found in [2].

Lemma 2.1.

Let X and Y be Hermitian matrices of order n and let $Z=X+Y$. Then

$$\begin{array}{c}{\lambda }_k(Z)\le {\lambda }_j(X)+{\lambda }_{k-j+1}(Y),n\ge k\ge j\ge 1,\\ {\lambda }_k(Z)\ge {\lambda }_j(X)+{\lambda }_{k-j+n}(Y),n\ge j\ge k\ge 1,\end{array}$$

where ${\lambda }_i(M)$ is the ith largest eigenvalue of the matrix M. In either of these inequalities, equality holds if and only if there exists a unit vector which is an eigenvector corresponding to each of the three eigenvalues involved.

The following lemma gives some basic properties of the α-adjacency matrix of G.

Lemma 2.2.

Let G be a connected graph of order n with m edges and having vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Then

1. ${\sum }_{i=1}^n{\rho }_i=2\alpha m$

2. ${\sum }_{i=1}^n{\rho }_i^2={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2$

3. ${\sum }_{i=1}^n{s}_i^2={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{4{\alpha }^2{m}^2}{n}.$

4. $\rho (G)\ge \frac{2m}{n}$, equality holds if and only if G is a degree regular graph.

5. $\rho (G)\ge \sqrt{\frac{Zg(G)}{n}}$, equality holds if and only if G is a degree regular graph.

Proof.

1. Clearly, ${\sum }_{i=1}^n{\rho }_i=\alpha {\sum }_{i=1}^n{d}_i+(1-\alpha ){\sum }_{i=1}^n{\lambda }_i=\alpha \sum _{u\in V(G)}{d}_G(u)=2\alpha m.$

2. Here, $$\begin{array}{c}\sum _{i=1}^n{\rho }_i^2=\sum _{i=1}^n{\left(\alpha d_i+(1-\alpha ){\lambda }_i\right)}^2={\alpha }^2\sum _{i=1}^n{d}_i^2+{(1-\alpha )}^2\sum _{i=1}^n{({\lambda }_i)}^2\\ ={\alpha }^2\sum _{u\in V(G)}{d}_G^2(u)+2{(1-\alpha )}^{2}m={\alpha }^2{Z}_g(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2.\end{array}$$

3. We have $$\sum _{i=1}^n{s}_i^2=\sum _{i=1}^n{\rho }_i^2-\frac{4{\alpha }^2{m}^2}{n},$$

and so by (2) the result follows.

• 4. Let $\mathbf{X}=\frac{1}{\sqrt{n}}(1,1,\cdots ,1)$ be a unit vector. Then, by Rayleigh-Ritz’s theorem for Hermitian matrices [2], we have $$\rho (G)\ge \frac{{\mathbf{X}}^t{A}_{\alpha }(G)\mathbf{X}}{{\mathbf{X}}^t\mathbf{X}}=\frac{\alpha \sum _{i=1}^n{d}_i+(1-\alpha )\sum _{i=1}^n{d}_i}{n}=\frac{2m}{n}.$$

Assume that G is k degree regular. Then each row sum of $A_{\alpha }(G)$ equals to a constant k. Therefore, by Perron-Frobenius theorem [2], k is a simple and largest eigenvalue of $A_{\alpha }(G).$ Thus $\rho (G)=k=\frac{nk}{n}=\frac{2m}{n}$ and equality holds. Conversely, assume that equality holds. Then $A_{\alpha }(G)\mathbf{X}=\rho (G)\mathbf{X}.$ Therefore, $d_i=\rho (G)$ for all i and thus G is degree regular.

• 5. This follows from [5]. Equality can be verified as in (4). □

From Case 3 of Lemma 2.2, we have ${\sum }_{i=1}^n{s}_i^2={(1-\alpha )}^2\parallel A(G){\parallel }_F^2+{\sum }_{i=1}^n{(\alpha d_i-\frac{2\alpha m}{n})}^2.$

Let (2.2) $$2S(G)={(1-\alpha )}^2\parallel A(G){\parallel }_F^2+\sum _{i=1}^n{\left(\alpha d_i-\frac{2\alpha m}{n}\right)}^2.$$(2.2) We observe that $2S(G)={(1-\alpha )}^2\parallel A(G){\parallel }_F^2$ if and only if G is $\frac{2m}{n}$-degree regular graph, otherwise $2S(G)>{(1-\alpha )}^2\parallel A(G){\parallel }_F^2.$ Further, $2S(G)=\parallel A(G)-\frac{2\alpha m}{n}{\mathbb{I}}_n{\parallel }_F^2={\sum }_{i=1}^n{s}_i^2,$ where ${\mathbb{I}}_n$ is the identity matrix of order $n.$

It is well known that a graph G has two distinct eigenvalues if and only if $G\cong K_n.$ Using this fact, it can be easily verified that the graph G has two distinct α-adjacency eigenvalues if and only if G is a complete graph with $\alpha \ne 1.$ The α-adjacency spectrum of the complete graph K_n is given in the next lemma [5].

Lemma 2.3.

If G = K_n is a complete graph, then the spectrum of $A_{\alpha }(K_n)$ is $\{(n-1),{(n\alpha -1)}^{[n-1]}\}$, where ${\rho }^{[j]}$ means the eigenvalues ρ is repeated j times in the spectrum.

The following lemma [5] gives a lower bound for the α-adjacency spectral radius.

Lemma 2.4.

If G is a graph with maximum degree $\Delta (G)=\Delta$, then

$$\rho (G)\ge \frac{1}{2}\left(\alpha (\Delta +1)+\sqrt{{\alpha }^2{(\Delta +1)}^2+4\Delta (1-2\alpha )}\right).$$

For $\alpha \in [0,1)$ and G being connected, equality holds if and only if $G\cong K_{1,\Delta }.$

We first find the α-adjacency energy of a degree regular graph.

Theorem 2.5.

If G is a degree regular graph of order n and $\alpha \in [0,1)$, then

$$E^{A_{\alpha }}(G)=(1-\alpha )E(G).$$

Proof.

Let ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$ be the adjacency eigenvalues of graph $G.$ If G is a k degree regular, then $D(G)=k{I}_n$ and so $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)=\alpha k{I}_n+(1-\alpha )A(G).$$ From this equality, it is clear that α-adjacency spectrum of G is $\{\alpha k+(1-\alpha ){\lambda }_1,\cdots ,\alpha k+(1-\alpha ){\lambda }_n\}.$ Using this and the fact $\frac{2\alpha m}{n}=\alpha k,$ we obtain $E^{A_{\alpha }}(G)=(1-\alpha )E(G).$ □

From Theorem 2.5, for a degree regular graph G, it is clear that the value of α-adjacency energy $E^{A_{\alpha }}(G)$ is a decreasing function of α, for $\alpha \in [0,1).$

The following theorem gives McClelland-type upper bound for α-adjacency energy in terms of order n and the quantity S(G) associated to G.

Theorem 2.6.

If G is a connected graph of order n and $\alpha \in [0,1)$, then $E^{A_{\alpha }}(G)\le \sqrt{2S(G)n}.$

Proof.

Using Cauchy-Schwarz’s inequality to the vectors $(|s_1|,|s_2|,\cdots ,|s_n|)$ and $(1,1,\cdots ,1),$ we have $${\left(E^{A_{\alpha }}(G)\right)}^2={\left(\sum _{i=1}^n\left|s_i\right|\right)}^2\le n\sum _{i=1}^n{s}_i^2=2nS(G)$$

Now, we obtain an upper bound for α-adjacency energy in terms of order n, size m and the quantity S(G) associated to G.

Theorem 2.7.

Let G be a connected graph of order $n\ge 3$ with m edges and having Zagreb index Zg(G). If $\alpha \in [0,\frac{1}{2}]$ or $\alpha \in (\frac{1}{2},1)$ and $Zg(G)>\frac{8m^2}{n}-2m$ or $Zg(G)<\frac{4m^2}{n}$, then

(2.3) $$E^{A_{\alpha }}(G)\le (1-\alpha )\left(\frac{2m}{n}\right)+\sqrt{(n-1)\left[2S(G)-{(1-\alpha )}^2{\left(\frac{2m}{n}\right)}^2\right]},$$(2.3)

where $2S(G)$ is same as in (2.2(2.2) $$2S(G)={(1-\alpha )}^2\parallel A(G){\parallel }_F^2+\sum _{i=1}^n{\left(\alpha d_i-\frac{2\alpha m}{n}\right)}^2.$$(2.2) ). Equality occurs if and only if either $G=K_n$ or G is a connected degree regular graph with three distinct eigenvalues given by $\frac{2m}{n},\frac{2m\alpha }{n}+(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}$ and $\frac{2m\alpha }{n}-(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}.$

Proof.

Let ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n$ be α-adjacency eigenvalues of G. For $1\le i\le n,$ let $s_i={\rho }_i(G)-\frac{2m\alpha }{n}.$ Using Lemma 2.2, we have ${\sum }_{i=2}^n{s}_i^2=2S(G)-s_1^2.$ Applying Cauchy-Schwarz’s inequality to the vectors $(|s_2|,|s_3|,\cdots ,|s_n|)$ and $(1,1,\cdots ,1),$ we obtain $$\sum _{i=2}^n\left|s_i\right|\le \sqrt{(n-1)\sum _{i=2}^n{s}_i^2}=\sqrt{(n-1)\left[2S(G)-s_1^2\right]}.$$ Therefore, we have $$E^{A_{\alpha }}(G)=s_1+\sum _{i=2}^n\left|s_i\right|\le s_1+\sqrt{(n-1)\left[2S(G)-s_1^2\right]}.$$ The last inequality suggests to consider the function $F(x)=x+\sqrt{(n-1)\left[2S(G)-x^2\right]}.$ It is easy to see that this function is strictly decreasing in the interval $\sqrt{2S(G)/n}<x\le \sqrt{2S(G)}.$ Since G is a connected graph, it follows that $m\ge n-1$ implying that $2m\ge 2n-2>n,$ for all $n\ge 3.$ We have $\sqrt{2S(G)/n}\le (1-\alpha )\frac{2m}{n}$ implying that (2.4) $$\gamma {\alpha }^2-2{\gamma }^{\prime }\alpha +{\gamma }^{\prime }\ge 0,$$(2.4) where $\gamma =\frac{8m^2}{n}-Zg(G)-2m$ and ${\gamma }^{\prime }=\frac{4m^2}{n}-2m.$ For α = 0, inequality (2.3) follows, as ${\gamma }^{\prime }=\frac{4m^2}{n}-2m>0.$ For $\alpha \in (0,1),$ consider the function $f(\alpha )=\gamma {\alpha }^2-2{\gamma }^{\prime }\alpha +{\gamma }^{\prime }.$ It is easy to see that $f(\alpha )$ is decreasing for $\alpha \le \frac{{\gamma }^{\prime }}{\gamma }$ and increasing for $\alpha \ge \frac{{\gamma }^{\prime }}{\gamma }.$ If $Zg(G)>\frac{8m^2}{n}-2m,$ then $\frac{{\gamma }^{\prime }}{\gamma }<0,$ as ${\gamma }^{\prime }>0$ and so $\frac{{\gamma }^{\prime }}{\gamma }\notin (0,1).$ This gives $f(\alpha )>f(0)={\gamma }^{\prime }>0$ and so inequality (2.4) follows in this case. So, assume that $Zg(G)\le \frac{8m^2}{n}-2m.$ Then $\frac{{\gamma }^{\prime }}{\gamma }>0.$ If $\frac{{\gamma }^{\prime }}{\gamma }\ge 1,$ then $Zg(G)\ge \frac{4m^2}{n}$ and so it follows that $f(\alpha )\ge f\left(\frac{1}{2}\right)=\frac{1}{4}\gamma >0,$ for all $\frac{4m^2}{n}\le Zg(G)\le \frac{8m^2}{n}-2m.$ So, if $\frac{4m^2}{n}\le Zg(G)\le \frac{8m^2}{n}-2m,$ then inequality (2.4) holds for all $\alpha \in (0,\frac{1}{2}].$ Now, assume that $Zg(G)<\frac{4m^2}{n}.$ Clearly, $\frac{{\gamma }^{\prime }}{\gamma }\in (0,1)$ and so we have $f\left(\frac{{\gamma }^{\prime }}{\gamma }\right)={\gamma }^{\prime }(1-\frac{{\gamma }^{\prime }}{\gamma })>0,$ as $\frac{{\gamma }^{\prime }}{\gamma }<1.$ So, if $Zg(G)<\frac{4m^2}{n},$ then inequality (2.3) holds for all $\alpha \in (0,1).$ Thus, it follows that the inequality $\sqrt{2S(G)/n}\le (1-\alpha )\frac{2m}{n}$ holds for all $\alpha \in [0,\frac{1}{2}]$ and holds for all $\alpha \in (\frac{1}{2},1),$ provided that $Zg(G)>\frac{8m^2}{n}-2m$ or $Zg(G)<\frac{4m^2}{n}.$ Since ${\rho }_1\ge \frac{2m}{n},$ that is, $(1-\alpha )\frac{2m}{n}\le s_1,$ therefore $\sqrt{2S(G)/n}\le (1-\alpha )\frac{2m}{n}\le s_1\le \sqrt{2S(G)},$ for all $\alpha \in [0,\frac{1}{2}]$ and for all $\alpha \in (\frac{1}{2},1),$ provided that $Zg(G)>\frac{8m^2}{n}-2m$ or $Zg(G)<\frac{4m^2}{n}.$ Now, F(x) being decreasing in $\sqrt{2S(G)/n}<x\le \sqrt{2S(G)},$ so $F(s_1)\le F((1-\alpha )\frac{2m}{n}).$ Thus, from this, inequality (2.3(2.3) $$E^{A_{\alpha }}(G)\le (1-\alpha )\left(\frac{2m}{n}\right)+\sqrt{(n-1)\left[2S(G)-{(1-\alpha )}^2{\left(\frac{2m}{n}\right)}^2\right]},$$(2.3) ) follows.

Assume that equality occurs in (2.3(2.3) $$E^{A_{\alpha }}(G)\le (1-\alpha )\left(\frac{2m}{n}\right)+\sqrt{(n-1)\left[2S(G)-{(1-\alpha )}^2{\left(\frac{2m}{n}\right)}^2\right]},$$(2.3) ). Then all the inequalities above occur as equalities. By Lemma 2.2, equality occurs in $(1-\alpha )\frac{2m}{n}\le s_1,$ if and only if G is a degree regular graph. Also, equality occurs in Cauchy-Schwarz’s inequality if $\left|s_2\right|=\left|s_3\right|=\cdots =\left|s_n\right|=\sqrt{\frac{2S(G)-{(1-\alpha )}^2{(\frac{2m}{n})}^2}{n-1}}.$ Since, $\sqrt{2S(G)/n}\le (1-\alpha )\frac{2m}{n}\le s_1$ holds for all $\alpha \in [0,\frac{1}{2}]$ and holds for all $\alpha \in (\frac{1}{2},1),$ provided that $Zg(G)>\frac{8m^2}{n}-2m$ or $Zg(G)<\frac{4m^2}{n},$ it follows that $s_1>\sqrt{\frac{2S(G)-{(1-\alpha )}^2{(\frac{2m}{n})}^2}{n-1}}.$ Thus there are two cases to consider. (i) Either G is a connected degree regular graph with two distinct α-adjacency eigenvalues (namely ${\rho }_1=\frac{2m}{n}$ and ${\rho }_2=\frac{2m\alpha }{n}-(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}$ repeated n – 1 times) or (ii) G is a connected degree regular graph with three distinct α-adjacency eigenvalues namely ${\rho }_1=\frac{2m}{n}$ and the other two given by $\frac{2m\alpha }{n}+(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}$ and $\frac{2m\alpha }{n}-(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}.$ In Case (i), by Lemma 2.3, it follows that G is a complete graph, that is, $G=K_n,$ while in Case (ii), it follows that G is a connected degree regular graph with three distinct eigenvalues given by $\frac{2m}{n},\frac{2m\alpha }{n}+(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}$ and $\frac{2m\alpha }{n}-(1-\alpha )\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}.$

Conversely, it can be easily verified that equality in (2.2) holds in each of above mentioned cases. □

Taking α = 0 and using the fact that $2S(G)=2m,$ we obtain the following result, which is the Koolen type [25] upper bound for the energy E(G).

Corollary 2.8.

Let G be a connected graph of order $n\ge 3$ with m edges. Then

$$E(G)\le \frac{2m}{n}+\sqrt{(n-1)\left[2m-{(\frac{2m}{n})}^2\right]}.$$

Equality occurs if and only if either $G=K_n$ or G is a degree regular graph with three distinct eigenvalues given by $\frac{2m}{n}$ and other two with absolute value $\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}.$

Taking $\alpha =\frac{1}{2}$ and using the fact that $2S(G)=\frac{1}{4}[2m+Zg(G)-\frac{4m^2}{n}]$ together with $2E^{A_{\frac{1}{2}}}(G)=QE(G),$ we obtain the following result, which is the Koolen type upper bound for the signless Laplacian energy QE(G).

Corollary 2.9.

Let G be a connected graph of order $n\ge 3$ with m edges having Zagreb index Zg(G). Then

$$QE(G)\le \frac{2m}{n}+\sqrt{(n-1)\left[2m+Zg(G)-\frac{4m^2}{n}\left(1+\frac{1}{n}\right)\right]}.$$

Equality occurs if and only if either $G=K_n$ or G is a degree regular graph with three distinct eigenvalues given by $\frac{4m}{n},\frac{2m}{n}+\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}$ and $\frac{2m}{n}+\sqrt{\frac{2m-{(\frac{2m}{n})}^2}{n-1}}.$

The following lemma gives a relation between α-adjacency eigenvalues of G and α-adjacency eigenvalues of spanning subgraphs of G.

Lemma 2.10.

Let G be a connected graph of order $n\ge 3$ and let $\alpha \in [\frac{1}{2},1)$. If $G^{\prime }$ is the graph obtained from G by deleting an edge, then for any $1\le i\le n$, we have ${\rho }_i(G)\ge {\rho }_i(G^{\prime }).$

Proof.

Let G be a connected graph of order $n\ge 3$ and let e = uv be an edge in G. Let $G^{\prime }=G-e$ be the graph obtained from G by deleting e. It is easy to see that (2.5) $$A_{\alpha }(G)=A_{\alpha }(G^{\prime })+N,$$(2.5) where N is the matrix of order n indexed by the vertices of G having ${(u,v)}^{th}$ and ${(v,u)}^{th}$ entries both equal to $1-\alpha ,$ and the ${(u,u)}^{th}$ and ${(v,v)}^{th}$ entries both equal to α, and all other entries equal to zero. It can be seen that the eigenvalues of the matrix N are $1^{[1]},2\alpha -1^{[1]},0^{[n-2]},$ where ${\lambda }^{[j]}$ means the eigenvalue λ is repeated j times in the spectrum. Taking $Z=A_{\alpha }(G),X=A_{\alpha }(G^{\prime }),Y=N$ and k = j = i in the second inequality of Lemma 2.1, we get ${\rho }_i(G)\ge {\rho }_i(G^{\prime }),$ provided that $\alpha \in [\frac{1}{2},1).$ □

In G, let $\eta =\eta (G)$ be the number of α-adjacency eigenvalues greater or equal to $\frac{2m\alpha }{n}.$ Since, by Lemma 2.2, we have ${\rho }_1\ge \frac{2m}{n},$ it follows that $1\le \eta \le n.$ Parameters similar to η have been considered for the graph matrices and therefore it will be interesting to connect the parameter η with α-adjacency energy of G. Next, we obtain an upper bound for α-adjacency energy in terms of order n, size m and the parameter η associated to G.

Theorem 2.11.

Let G be a connected graph of order $n\ge 3$ and let $\frac{1}{2}\le \alpha <1.$ Then $$E^{A_{\alpha }}(G)\le 2(n-1)+2(\eta -1)(\alpha n-1)-\frac{4\alpha \eta m}{n},$$ with equality if and only if $G\cong K_n.$

Proof.

Let G be a connected graph of order n having α-adjacency eigenvalues ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n.$ Let η be the positive integer such that ${\rho }_{\eta }\ge \frac{2\alpha m}{n}$ and ${\rho }_{\eta +1}<\frac{2\alpha m}{n}.$ Using (1) of Lemma 2.2 and the definition of α-adjacency energy, we have $$\begin{array}{l}{E}^{A\alpha }(G)=\sum _{i=1}^{\eta }\left|{\rho }_i-\frac{2\alpha m}{n}\right|=\sum _{i=1}^{\eta }\left({\rho }_i-\frac{2\alpha m}{n}\right)+\sum _{i=\eta +1}^n\left(\frac{2\alpha m}{n}-{\rho }_i\right)\\ =2\left(\sum _{i=1}^n{\rho }_i-\frac{2\eta \alpha m}{n}\right).\end{array}$$ Clearly G is a spanning subgraph of K_n. So, from Lemma 2.10, it follows that ${\rho }_i(G)\le {\rho }_i(K_n)$ for each $1\le i\le n.$ Therefore, we have (2.6) $$\sum _{i=1}^{\eta }{\rho }_i(G)\le \sum _{i=1}^{\eta }{\rho }_i(K_n)=n-1+(\eta -1)(\alpha n-1).$$(2.6) Using (2.6(2.6) $$\sum _{i=1}^{\eta }{\rho }_i(G)\le \sum _{i=1}^{\eta }{\rho }_i(K_n)=n-1+(\eta -1)(\alpha n-1).$$(2.6) ), we obtain $$E^{A_{\alpha }}(G)\le 2(n-1)+2(\eta -1)(\alpha n-1)-\frac{4\alpha \eta m}{n}.$$ Assume that equality occurs so that equality occurs in (2.6(2.6) $$\sum _{i=1}^{\eta }{\rho }_i(G)\le \sum _{i=1}^{\eta }{\rho }_i(K_n)=n-1+(\eta -1)(\alpha n-1).$$(2.6) ). Since, equality occurs in (2.6(2.6) $$\sum _{i=1}^{\eta }{\rho }_i(G)\le \sum _{i=1}^{\eta }{\rho }_i(K_n)=n-1+(\eta -1)(\alpha n-1).$$(2.6) ) if and only if $G\cong K_n,$ it follows that equality holds if and only if $G\cong K_n.$ □

The following lemma will be required in the sequel.

Lemma 2.12.

Let G be a connected graph of order n, size m and having vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Then

$$\rho (G)\ge \sqrt{\frac{Zg(G)}{n}}\ge \frac{2m}{n}.$$

Proof.

The first inequality follows by Case 5 of Lemma 2.2. Therefore, we need to prove the second inequality. Applying Cauchy-Schwartz inequality to ${({\sum }_{i=1}^n{d}_i)}^2,$ we have ${({\sum }_{i=1}^n{d}_i)}^2\le n{\sum }_{i=1}^n{d}_i^2,$ which implies ${\sum }_{i=1}^n{d}_i^2\ge \frac{{(2m)}^2}{n}$ and hence $\sqrt{{\sum }_{i=1}^n{d}_i^2}\ge \frac{2m}{\sqrt{n}}.$ Thus, $$\sqrt{\frac{Zg(G)}{n}}=\sqrt{\frac{\sum _{i=1}^n{d}_i^2}{n}}\ge \frac{2m}{n}.$$ This completes the proof. □

The following theorems give upper bounds for the α-adjacency energy in terms of order n, size m, Zagreb index Zg(G) and the parameter α.

Theorem 2.13.

Let G be a connected graph of order $n\ge 3$ having Zagreb index Zg(G) and let $\alpha \le 1-\frac{n}{2m}.$ Then

(2.7) $$\begin{array}{cc}{E}^{A_{\alpha }}(G)\hfill & \le {\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{2\alpha m}{n^2}(2\alpha nm+2\alpha m+n)+\hspace{0.17em}\text{ln}\hspace{0.17em}\left(\frac{\theta }{\Gamma }\right)\hfill \\ \hfill & \hspace{1em}+\frac{4\alpha m}{n}\left(\sqrt{\frac{Zg(G)}{n}}\right)-\left(\sqrt{\frac{Zg(G)}{n}}\right)\left(\sqrt{\frac{Zg(G)}{n}}-1\right),\hfill \end{array}$$(2.7)

where $\Gamma =\left|\mathit{det}(A_{\alpha }(G)-\frac{2\alpha m}{n}{\mathbb{I}}_n)\right|$ and $\theta =\sqrt{\frac{Zg(G)}{n}}-\frac{2\alpha m}{n}.$ Equality holds if and only if $G\cong K_n$ and α = 0 or G is a k-degree regular graph with three distinct α-adjacency eigenvalues given by $k,\alpha k+1$ and $\alpha k-1.$

Proof.

Let G be a connected graph of order n and let ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n$ be the α-adjacency eigenvalues of G. Consider the function $$f(x)={\left(x-\frac{2\alpha m}{n}\right)}^2-\left(x-\frac{2\alpha m}{n}\right)-\hspace{0.17em}\text{ln}\hspace{0.17em}\left(x-\frac{2\alpha m}{n}\right),\left(x-\frac{2\alpha m}{n}\right)>0.$$ It is easy to see that this function is non-decreasing for $x-\frac{2\alpha m}{n}\ge 1$ and non-increasing for $0\le \left(x-\frac{2\alpha m}{n}\right)\le 1.$ So, we have $f(x)\ge f\left(\frac{2\alpha m}{n}+1\right)=0$ implying that $$\left(x-\frac{2\alpha m}{n}\right)\le {\left(x-\frac{2\alpha m}{n}\right)}^2-\hspace{0.17em}\text{ln}\hspace{0.17em}\left(x-\frac{2\alpha m}{n}\right)$$ for $\left(x-\frac{2\alpha m}{n}\right)>0,$ with equality if and only if $\left(x-\frac{2\alpha m}{n}\right)=1.$ Using these observations in the definition of α-adjacency energy, we have (2.8) $$\begin{array}{l}{E}^{A\alpha }(G)={\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|\\ \le {\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left({\left({\rho }_i-\frac{2\alpha m}{n}\right)}^2-\hspace{0.17em}\text{ln}\hspace{0.17em}\left|{\rho }_i-\frac{2\alpha m}{n}\right|\right)\\ ={\rho }_1-\frac{2\alpha m}{n}+{\alpha }^{2}Zg(G)+{(1-\alpha )}^2{\Vert A(G)\Vert }_F^2-\left(\frac{4{\alpha }^2{m}^2}{n^2}\right)(n-1)-{\rho }_1^2\\ -\hspace{0.17em}\text{ln}\hspace{0.17em}\prod _{i=1}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|+\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right)-\frac{4\alpha m}{n}(2\alpha m-{\rho }_1)\\ ={\alpha }^{2}Zg(G)+{(1-\alpha )}^2{\Vert A(G)\Vert }_F^2-\frac{2\alpha m}{n}(2\alpha nm+2\alpha m+n)+\frac{4\alpha m}{n}{\rho }_1\\ -\hspace{0.17em}\text{ln}\hspace{0.17em}\Gamma +\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right)-{\rho }_1({\rho }_1-1).\end{array}$$(2.8) Consider the function $$\begin{array}{c}g(x)={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2\\ -\frac{2\alpha m}{n^2}(2\alpha nm+2\alpha m+n)+\frac{4\alpha m}{n}x\\ \hspace{1em}-\hspace{0.17em}\text{ln}\hspace{0.17em}\Gamma +\hspace{0.17em}\text{ln}\hspace{0.17em}\left(x-\frac{2\alpha m}{n}\right)-x(x-1).\end{array}$$ Evidently, the function g(x) is increasing for $0\le x-\frac{2\alpha m}{n}\le 1$ and decreasing for $x-\frac{2\alpha m}{n}\ge 1.$ Since $x-\frac{2\alpha m}{n}\ge (1-\alpha )\frac{2m}{n}\ge 1$ provided that $\alpha \le 1-\frac{n}{2m},$ then t for $\alpha \le 1-\frac{n}{2m},$ it follows that $x-\frac{2\alpha m}{n}\ge 1.$ Further, $(1-\alpha )\frac{2m}{n}\ge 1$ implies that $\frac{2m}{n}\ge 1+\frac{2m\alpha }{n}$ and by Lemma 2.12, we have $x\ge \sqrt{\frac{Zg(G)}{n}}\ge \frac{2m}{n}.$ Therefore, it follows that (2.9) $$\begin{array}{l}g(x)\le g\left(\sqrt{\frac{Zg(G)}{n}}\right)\\ ={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{2\alpha m}{n^2}(2\alpha nm+2\alpha m+n)\\ \hspace{1em}+\frac{4\alpha m}{n}\sqrt{\frac{Zg(G)}{n}}+\hspace{0.17em}\text{ln}\hspace{0.17em}\left(\frac{\theta }{\Gamma }\right)-\frac{Zg(G)}{n}-\sqrt{\frac{Zg(G)}{n}}.\end{array}$$(2.9) Combining inequalities (2.9(2.9) $$\begin{array}{l}g(x)\le g\left(\sqrt{\frac{Zg(G)}{n}}\right)\\ ={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{2\alpha m}{n^2}(2\alpha nm+2\alpha m+n)\\ \hspace{1em}+\frac{4\alpha m}{n}\sqrt{\frac{Zg(G)}{n}}+\hspace{0.17em}\text{ln}\hspace{0.17em}\left(\frac{\theta }{\Gamma }\right)-\frac{Zg(G)}{n}-\sqrt{\frac{Zg(G)}{n}}.\end{array}$$(2.9) ) and (2.8$$\begin{array}{c}g(x)={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2\\ -\frac{2\alpha m}{n^2}(2\alpha nm+2\alpha m+n)+\frac{4\alpha m}{n}x\\ \hspace{1em}-\hspace{0.17em}\text{ln}\hspace{0.17em}\Gamma +\hspace{0.17em}\text{ln}\hspace{0.17em}\left(x-\frac{2\alpha m}{n}\right)-x(x-1).\end{array}$$), we arrive at (2.7(2.8) $$\begin{array}{l}{E}^{A\alpha }(G)={\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|\\ \le {\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left({\left({\rho }_i-\frac{2\alpha m}{n}\right)}^2-\hspace{0.17em}\text{ln}\hspace{0.17em}\left|{\rho }_i-\frac{2\alpha m}{n}\right|\right)\\ ={\rho }_1-\frac{2\alpha m}{n}+{\alpha }^{2}Zg(G)+{(1-\alpha )}^2{\Vert A(G)\Vert }_F^2-\left(\frac{4{\alpha }^2{m}^2}{n^2}\right)(n-1)-{\rho }_1^2\\ -\hspace{0.17em}\text{ln}\hspace{0.17em}\prod _{i=1}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|+\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right)-\frac{4\alpha m}{n}(2\alpha m-{\rho }_1)\\ ={\alpha }^{2}Zg(G)+{(1-\alpha )}^2{\Vert A(G)\Vert }_F^2-\frac{2\alpha m}{n}(2\alpha nm+2\alpha m+n)+\frac{4\alpha m}{n}{\rho }_1\\ -\hspace{0.17em}\text{ln}\hspace{0.17em}\Gamma +\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right)-{\rho }_1({\rho }_1-1).\end{array}$$(2.8) ).

Assume that inequality holds in (2.7(2.8) $$\begin{array}{l}{E}^{A\alpha }(G)={\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|\\ \le {\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left({\left({\rho }_i-\frac{2\alpha m}{n}\right)}^2-\hspace{0.17em}\text{ln}\hspace{0.17em}\left|{\rho }_i-\frac{2\alpha m}{n}\right|\right)\\ ={\rho }_1-\frac{2\alpha m}{n}+{\alpha }^{2}Zg(G)+{(1-\alpha )}^2{\Vert A(G)\Vert }_F^2-\left(\frac{4{\alpha }^2{m}^2}{n^2}\right)(n-1)-{\rho }_1^2\\ -\hspace{0.17em}\text{ln}\hspace{0.17em}\prod _{i=1}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|+\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right)-\frac{4\alpha m}{n}(2\alpha m-{\rho }_1)\\ ={\alpha }^{2}Zg(G)+{(1-\alpha )}^2{\Vert A(G)\Vert }_F^2-\frac{2\alpha m}{n}(2\alpha nm+2\alpha m+n)+\frac{4\alpha m}{n}{\rho }_1\\ -\hspace{0.17em}\text{ln}\hspace{0.17em}\Gamma +\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right)-{\rho }_1({\rho }_1-1).\end{array}$$(2.8) ). Then all the inequalities above occur as equalities. By Lemma 2.2, equality occurs in ${\rho }_1\ge \sqrt{\frac{Zg(G)}{n}}$ if and only if G is a degree regular graph. For equality in (2.8$$\begin{array}{c}g(x)={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2\\ -\frac{2\alpha m}{n^2}(2\alpha nm+2\alpha m+n)+\frac{4\alpha m}{n}x\\ \hspace{1em}-\hspace{0.17em}\text{ln}\hspace{0.17em}\Gamma +\hspace{0.17em}\text{ln}\hspace{0.17em}\left(x-\frac{2\alpha m}{n}\right)-x(x-1).\end{array}$$), we have $\left|{\rho }_2-\frac{2\alpha m}{n}\right|=\cdots =\left|{\rho }_n-\frac{2\alpha m}{n}\right|=1.$ For $i=2,3,\cdots ,n,$ the quantity $\left|{\rho }_i-\frac{2\alpha m}{n}\right|$ can have at most two distinct values and therefore we have the following cases.

• Case 1. For all $i=2,3,\cdots ,n,$ if ${\rho }_i-\frac{2\alpha m}{n}=1,$ then ${\rho }_i=1+\frac{2\alpha m}{n},$ implying that G has two distinct α-adjacency eigenvalues, namely ${\rho }_1=\frac{2m}{n}$ and ${\rho }_i=1+\frac{2\alpha m}{n}.$ So, by Lemma 2.3, equality occurs for the complete graph K_n, provided that α-adjacency eigenvalues of K_n are n – 1 with multiplicity 1 and $\alpha n-1$ with multiplicity n – 1. It is clear that equality can not hold in this case.

• Case 2. For all $i=2,3,\cdots ,n,$ if ${\rho }_i-\frac{2\alpha m}{n}=-1,$ then ${\rho }_i=\frac{2\alpha m}{n}-1,$ implying that G has two distinct α-adjacency eigenvalues, namely ${\rho }_1=\frac{2m}{n}$ and ${\rho }_i=\frac{2\alpha m}{n}-1.$ So, using Lemma 2.3, it follows that equality occurs for the complete graph K_n, provided that α = 0.

• Case 3. For the remaining case, for some t, let ${\rho }_i-\frac{2\alpha m}{n}=1,$ for $i=2,3,\cdots ,t,$ and ${\rho }_i-\frac{2\alpha m}{n}=-1,$ for $i=t+1,\cdots ,n.$ This implies that G is degree regular graph with three distinct α-adjacency eigenvalues, namely ${\rho }_1=\frac{2m}{n}$ with multiplicity 1, ${\rho }_i=1+\alpha {\rho }_1$ with multiplicity t – 1 and ${\rho }_i=\alpha {\rho }_1-1$ with multiplicity n – t.

Conversely, if $G\cong K_n,$ then ${\rho }_1=n-1,{\rho }_i=\alpha n-1,$ for $i=2,3,\cdots ,n$ and $\frac{2\alpha m}{n}=\alpha (n-1).$ It can be seen that equality occurs in (2.6). On the other hand, if G is a degree regular graph with three distinct α-adjacency eigenvalues, namely ${\rho }_1,\alpha {\rho }_1+1$ and $\alpha {\rho }_1-1,$ then from the above discussion, it is clear that the equality holds in (2.7). □

The following result gives another upper bound for the α-adjacency energy of a graph G in terms of order n, size m and the Zagreb index Zg(G).

Theorem 2.14.

Let G be a connected graph of order $n\ge 3$ having Zagreb index Zg(G) and let $\alpha \le 1-\frac{n}{2m}.$ Then

$$\begin{array}{cc}{E}^{A_{\alpha }}(G)\hfill & \le {\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2+\hspace{0.17em}\text{ln}\hspace{0.17em}\left(\frac{2m(1-\alpha )}{n\Gamma }\right)\hfill \\ \hfill & \hspace{1em}-\frac{2\alpha m}{n^2}\left(2n\alpha m+2\alpha m-4m+n\right)-\frac{2m}{n^2}\left(2m-n\right),\hfill \end{array}$$

where $\Gamma =\left|\mathit{det}(A_{\alpha }(G)-\frac{2\alpha m}{n}{\mathbb{I}}_n)\right|.$ Equality holds if and only if $G\cong K_n$ and α = 0 or G is a k-degree regular graph with three distinct α-adjacency eigenvalues given by $k,\alpha k+1$ and $\alpha k-1.$

Proof.

The proof is similar to the proof of Theorem 2.13.

3. Lower bounds for the α-adjacency energy of graphs

The following theorem gives a lower bound for the α-adjacency energy in terms of order n, size m and the Zagreb index Zg(G).

Theorem 3.1.

If G is a connected graph of order $n\ge 3$, size m and Zagreb index Zg(G), then

(3.10) $$E^{A_{\alpha }}(G)\ge \sqrt{2\left({\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{2{(\alpha m)}^2}{n}\right)}.$$(3.10)

for $\alpha \in [0,1)$

Proof.

Let $s_1\ge s_2\ge \cdots \ge s_n$ be the auxiliary eigenvalues as defined earlier. We have $${\left(E^{A_{\alpha }}(G)\right)}^2={\left(\sum _{i=1}^n\left|s_i\right|\right)}^2=\sum _{i=n}^n{s}_i^2(G)+2\sum _{1\le i<j\le n}\left|s_i{s}_j\right|.$$ By Case 3 of Lemma 2.2, we have (3.11) $$\sum _{i=1}^n{s}_i^2(G)={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{4{\alpha }^2{m}^2}{n}.$$(3.11) Also, $$\begin{array}{l}2\sum _{1\le i<j\le n}\left|s_i{s}_j\right|\ge 2\left|\sum _{1\le i<j\le n}\left({\rho }_i(G)-\frac{2\alpha m}{n}\right)\left({\rho }_j(G)-\frac{2\alpha m}{n}\right)\right|\\ =2\left|\sum _{1\le i<j\le n}{\rho }_i(G){\rho }_j(G)-\frac{4{\alpha }^2{m}^2}{n}(n-1)+\frac{4{\alpha }^2{m}^2}{n^2}n(n-1)\right|\\ =2\left|\sum _{i=1}^n{\rho }_i(G){\rho }_j(G)\right|=\sum _{i=1}^n{\rho }_i^2(G)={\alpha }^2\sum _{i=1}^n{d}_i^2+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2.\end{array}$$ Using this inequality and (3.11(3.11) $$\sum _{i=1}^n{s}_i^2(G)={\alpha }^{2}Zg(G)+{(1-\alpha )}^2\parallel A(G){\parallel }_F^2-\frac{4{\alpha }^2{m}^2}{n}.$$(3.11) ), clearly (3.10$${\left(E^{A_{\alpha }}(G)\right)}^2={\left(\sum _{i=1}^n\left|s_i\right|\right)}^2=\sum _{i=n}^n{s}_i^2(G)+2\sum _{1\le i<j\le n}\left|s_i{s}_j\right|.$$) follows. □

Now, we obtain a lower bound for the α-adjacency energy in terms of order n, size m and the parameter α.

Theorem 3.2.

Let G be a connected graph of order $n\ge 3$ and size m and let $\alpha \in (0,1)$. Then

(3.12) $$E^{A_{\alpha }}(G)\ge 4(1-\alpha )\frac{m}{n}.$$(3.12)

Equality occurs if and only if G is degree regular with one positive and n –1 negative adjacency eigenvalues.

Proof.

Let G be a connected graph of order n and having α-adjacency eigenvalues ${\rho }_1\ge {\rho }_2\ge \cdots \ge {\rho }_n.$ Let η be a positive integer such that ${\rho }_{\eta }\ge \frac{2\alpha m}{n}$ and ${\rho }_{\eta +1}<\frac{2\alpha m}{n}.$ Using Case 1 of Lemma 2.2 and the definition of α-adjacency energy, we have $$\begin{array}{l}{E}^{A_{\alpha }}(G)=\sum _{i=1}^n\left|{\rho }_i-\frac{2\alpha W(G)}{n}\right|=\sum _{i=1}^{\eta }\left({\rho }_i-\frac{2\alpha m}{n}\right)+\sum _{i=\eta +1}^n\left(\frac{2\alpha m}{n}-{\rho }_i\right)\\ =2\left(\sum _{i=1}^{\eta }{\rho }_i-\frac{2\eta \alpha m}{n}\right).\end{array}$$ First we show that (3.13) $$E^{A_{\alpha }}(G)=2\left(\sum _{i=1}^{\eta }{\rho }_i-\frac{2\eta \alpha m}{n}\right)=2\underset{1\le j\le n}{\max }\left\{\sum _{i=1}^j{\rho }_i-\frac{2\alpha jm}{n}\right\}.$$(3.13) Since $1\le \eta \le n,$ it follows that either $\eta <j$ or $\eta \ge j.$ If $j>\eta ,$ then we have $$\begin{array}{c}\sum _{i=1}^j{\rho }_i-\frac{2\alpha jm}{n}=\sum _{i=1}^{\eta }{\rho }_i+\sum _{i=\eta +1}^j{\rho }_i-\frac{2\alpha jm}{n}\\ <\sum _{i=1}^{\eta }{\rho }_i-\frac{2\alpha \eta m}{n}\end{array}$$ as ${\rho }_i<\frac{2\alpha m}{n},$ for $i\ge \eta +1.$ Similarly, for $j\le \eta ,$ it can be seen that $$\sum _{i=1}^j{\rho }_i-\frac{2\alpha jm}{n}\le \sum _{i=1}^{\eta }{\rho }_i-\frac{2\alpha \eta m}{n}.$$ This proves (3.13(3.13) $$E^{A_{\alpha }}(G)=2\left(\sum _{i=1}^{\eta }{\rho }_i-\frac{2\eta \alpha m}{n}\right)=2\underset{1\le j\le n}{\max }\left\{\sum _{i=1}^j{\rho }_i-\frac{2\alpha jm}{n}\right\}.$$(3.13) ). Therefore, we have $$\begin{array}{c}{E}^{A_{\alpha }}(G)=2\underset{1\le j\le n}{\max }\left\{\sum _{i=1}^j{\rho }_i-\frac{4\alpha jm}{n}\right\}\ge 2{\rho }_1-\frac{4\alpha m}{n}\\ \ge \frac{4m}{n}-\frac{4\alpha m}{n}=(1-\alpha )\frac{4m}{n}.\end{array}$$ Suppose equality holds in (3.12). Then all the inequalities above occur as equalities. By Lemma 2.2, equality occurs in ${\rho }_1\ge \frac{2m}{n}$ if and only if G is a degree regular graph. Also, equality occurs in $2{\max }_{1\le j\le n}\left\{{\sum }_{i=1}^j{\rho }_i-\frac{4\alpha jm}{n}\right\}\ge 2{\rho }_1-\frac{4\alpha m}{n}$ if and only if η = 1. Thus, it follows that equality occurs in (3.12) if and only if G is a degree regular graph with η = 1. Let G be a k-degree regular graph having adjacency eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n.$ Then, by Theorem 2.5, we have ${\rho }_1=k$ and ${\rho }_i=\alpha k+(1-\alpha ){\lambda }_i,$ for $i=2,3,\cdots ,n.$ Since η = 1, for $2\le i\le n,$ we have ${\rho }_i<\frac{2\alpha m}{n}=\alpha k,$ which gives $\alpha k+(1-\alpha ){\lambda }_i<\alpha k,$ which further gives ${\lambda }_i<0$ as $1-\alpha >0.$ Thus, it follows that equality occurs in (3.12) if and only if G is a degree regular graph with one positive and n – 1 negative adjacency eigenvalues. □

Proceeding similarly as in Theorem 3.2 and using Case 5 of Lemma 2.2, we obtain the following lower bound for α-adjacency energy of a connected graph.

Theorem 3.3.

Let G be a connected graph of order $n\ge 3$, size m and Zagreb index Zg(G). For $\alpha \in (0,1)$, we have

$$E^{A_{\alpha }}(G)\ge 2\sqrt{\frac{Zg(G)}{n}}-\frac{4\alpha m}{n}.$$

Equality occurs if and only if G is degree regular with one positive and n – 1 negative adjacency eigenvalues.

For a graph G of order $n\ge 2,$ it is well known that G has one positive adjacency eigenvalue if and only if G is a complete k-partite graph, where $2\le k\le n-1.$ Therefore, if α = 0, then following the proof of Theorem 3.2, we get $$E(G)=E^{A_0}(G)\ge 2{\lambda }_1\ge \frac{4m}{n}.$$ Equality occurs if and only if G is a regular graph with one positive adjacency eigenvalue. That is, equality occurs if and only if G is a regular complete k-partite graph, where $2\le k\le n-1.$

The following theorem gives a lower bound for α-adjacency energy of a connected graph in terms of order n, size m, the maximum degree Δ and the parameter α.

Theorem 3.4.

Let G be a connected graph of order $n\ge 3$, size m and maximum degree $\Delta .$ For $\alpha \in [0,1)$, we have

(3.14) $$E^{A_{\alpha }}(G)\ge \left(\alpha (△+1)+\sqrt{{\alpha }^2{(△+1)}^2+4△(1-2\alpha )}\right)-\frac{4\alpha m}{n},$$(3.14)

with equality if and only if $G\cong K_{1,\Delta }.$

Proof.

By Eq. (3.13)(3.13) $$E^{A_{\alpha }}(G)=2\left(\sum _{i=1}^{\eta }{\rho }_i-\frac{2\eta \alpha m}{n}\right)=2\underset{1\le j\le n}{\max }\left\{\sum _{i=1}^j{\rho }_i-\frac{2\alpha jm}{n}\right\}.$$(3.13) and Lemma 2.4, we have $$\begin{array}{c}{E}^{A_{\alpha }}(G)=\underset{1\le j\le n}{\max }\left\{2\sum _{i=1}^j{\rho }_i-\frac{4\alpha im}{n}\right\}\ge 2{\rho }_1(G)-\frac{4\alpha m}{n}\\ \ge \alpha (\Delta +1)+\sqrt{{\alpha }^2{(\Delta +1)}^2+4\Delta (1-2\alpha )}-\frac{4\alpha m}{n}.\end{array}$$ Suppose equality holds in (3.13). Then all the inequalities above occur as equalities. Since equality occurs in Lemma 2.4 if and only if $G\cong K_{1,\Delta }$ and equality occurs in $$\underset{1\le j\le n}{\max }\left\{2\sum _{i=1}^j{\rho }_i-\frac{4\alpha im}{n}\right\}\ge 2{\rho }_1(G)-\frac{4\alpha m}{n}$$ if and only if η = 1, it follows that equality occurs in (3.14) if and only if $G\cong K_{1,\Delta },\Delta =n-1$ and η = 1. For the graph $K_{1,\Delta },$ the α-adjacency eigenvalues are $\left\{{\alpha }^{[n-2]},\frac{\alpha (\Delta +1)\pm \sqrt{D}}{2}\right\},$ where $D={\alpha }^2{(\Delta +1)}^2+4\Delta (1-2\alpha )$ and average of the α-adjacency equals to $2\alpha -\frac{2\alpha }{n}.$ Clearly, now η = 1 for $K_{1,\Delta }.$ Thus, equality occurs in (3.14) if and if $G\cong K_{1,\Delta }.$ This completes the proof. □

Now, we obtain a lower bound for α-adjacency energy of a connected graph in terms of order n, size m and Zagreb index Zg(G).

Theorem 3.5.

Let G be a connected graph of order $n\ge 3$ having size m and Zagreb index Zg(G). For $\alpha \in [0,1)$, we have

(3.15) $$E^{A_{\alpha }}(G)\ge \sqrt{\frac{Zg(G)}{n}}+(n-1)+\hspace{0.17em}\text{ln}\hspace{0.17em}\left(\frac{\Gamma }{\theta }\right),$$(3.15)

where $\Gamma =\left|\mathit{det}(A_{\alpha }(G)-\frac{2\alpha m}{n}{\mathbb{I}}_n)\right|$ and $\theta =\sqrt{\frac{Zg(G)}{n}}-\frac{2m\alpha }{n}$. Equality holds as in Theorem 2.13.

Proof.

Consider the function $f(x)=x-1-\hspace{0.17em}\text{ln}\hspace{0.17em}x,$ where $x>0.$ It is easy to verify that the function f(x) is increasing for $x\ge 1$ and decreasing for $0\le x\le 1.$ Therefore, we have $f(x)\ge f(1)=0$ implying that $x\ge 1+\hspace{0.17em}\text{ln}\hspace{0.17em}x,$ for $x>0,$ with equality if and only if $x=1.$ Using this observation with $x=|{\rho }_i-\frac{2\alpha m}{n}|,$ for $2\le i\le n$ and the definition of α-adjacency energy, we have $$\begin{array}{l}{E}^{A_{\alpha }}(G)={\rho }_1-\frac{2\alpha m}{n}+\sum _{i=2}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|\\ \ge {\rho }_1-\frac{2\alpha m}{n}+(n-1)+\sum _{i=2}^n\hspace{0.17em}\text{ln}\hspace{0.17em}\left|{\rho }_i-\frac{2\alpha m}{n}\right|\\ ={\rho }_1-\frac{2\alpha m}{n}+(n-1)+\hspace{0.17em}\text{ln}\hspace{0.17em}\prod _{i=2}^n\left|{\rho }_i-\frac{2\alpha m}{n}\right|\\ ={\rho }_1-\frac{2\alpha m}{n}+(n-1)+\hspace{0.17em}\text{ln}\hspace{0.17em}\left|\mathit{det}\left(A_{\alpha }(G)-\frac{2\alpha m}{n}\right)\right|-\hspace{0.17em}\text{ln}\hspace{0.17em}\left({\rho }_1-\frac{2\alpha m}{n}\right).\end{array}$$ Now, consider the function $g(x)=x-\frac{2\alpha m}{n}+(n-1)+\hspace{0.17em}\text{ln}\hspace{0.17em}\left|\mathit{det}(A_{\alpha }(G)-\frac{2\alpha m}{n})\right|-\hspace{0.17em}\text{ln}\hspace{0.17em}\left(x-\frac{2\alpha m}{n}\right).$ Clearly, g(x) is increasing for $x-\frac{2\alpha m}{n}\ge 1.$ Since, $x-\frac{2\alpha m}{n}\ge (1-\alpha )\frac{2m}{n}\ge 1$ implying that $\alpha \le 1-\frac{n}{2m},$ therefore, for $\alpha \le 1-\frac{n}{2m},$ it follows that $x-\frac{2\alpha m}{n}\ge 1.$ Further, $(1-\alpha )\frac{2m}{n}\ge 1$ implies that $\frac{2m}{n}\ge 1+\frac{2m\alpha }{n}.$ From Lemma 2.2 and the fact that g(x) is increasing for $1+\frac{2m\alpha }{n}\le \frac{2m}{n}\le \sqrt{\frac{Zg(G)}{n}}\le x,$ it follows that $g(x)\ge g(\sqrt{\frac{Zg(G)}{n}}).$ From this, inequality (3.15) follows. Equality case can be discussed similar to Theorem 2.13. □

A lower bound similar to the lower bound given in the above theorem can be obtained for $\rho \ge \frac{2m}{n}.$

4. Conclusion

Let ${\mathbb{M}}_n(\mathbb{C})$ be the set of all square matrices of order n with complex entries. The trace norm of a matrix $M\in {\mathbb{M}}_n(\mathbb{C})$ is defined as $\Vert M{\Vert }_{*}={\sum }_{i=1}^n{\sigma }_i(M),$ where ${\sigma }_1(M)\ge {\sigma }_2(M)\ge \cdots \ge {\sigma }_n(M)$ are the singular values of M. It is well known that for a Hermitian matrix M, if ${\sigma }_i(M)$ are the singular values and ${\lambda }_i(M),i=1,2,\cdots ,n,$ are the eigenvalues, then ${\sigma }_i(M)=|{\lambda }_i(M)|.$ In the light of this definition, it follows that the α-adjacency energy $E^{A_{\alpha }}(G)$ of a connected graph G introduced in Sec. 1, is the trace norm of the matrix $A_{\alpha }(G)-\frac{2\alpha m}{n}{I}_n.$ It is an interesting problem in Matrix theory to determine among a given class of matrices the matrix (or the matrices) which attain the maximum value and the minimum value for the trace norm. The trace norm of matrices associated with the graphs and digraphs are extensively studied. Various papers can be found in the literature in this direction. This gives another motivation for the study of α-adjacency energy of a graph G.

In this paper we have explored some properties of α-adjacency energy, particularly we have focused on the estimates for this spectral graph invariant connecting it with different graph parameters associated with the structure of the graph. It will be of interest in future to explore some more structure related properties. Moreover, it will be an interesting problem to find extremal graphs for this quantity among a class of graphs with given one or more parameters. Further, as is clear from Theorem 3.2 that the α-adjacency energy of a graph G is closely related with the parameter $S_{\alpha ,k}(G)={\sum }_{i=1}^k{\rho }_i(G),$ which represents the Ky Fan k-norm of the matrix $A_{\alpha }(G).$ This is another direction which will be great interest to study the spectral graph invariant $S_{\alpha ,k}(G)$ and obtain extremal results for it.

Additional information

Funding

The research of S. Pirzada was supported by SERB-DST, New Delhi under the research project number MTR/2017/000084.