The eccentric harmonic index of trees

Abstract

Topological indices play an important role in mathematical chemistry, particularly in studies of quantitative structure property and quantitative structure activity relationships. The eccentric harmonic index of G is defined as $H_e(G)=\sum _{uv\in E(G)}\frac{2}{e(u)+e(v)},$ where uv is an edge of G, e(u) is the eccentricity of the vertex u in G. In this paper, we will determine the maximum and minmum eccentric harmonic index of trees in terms of graph parameters such as pendant number and matching number, and characterize corresponding extremal graphs.

Keywords:

• The eccentric harmonic index
• matching number
• pendant vertices
• trees

MATHEMATICS SUBJECT CLASSIFICATION:

• 05C05
• 05C35
• 05C75

1 Introduction

Let G be a simple graph with vertex set V(G) and edge set E(G). Its order is $|V(G)|$. Let d(v) and N(v) be the degree and the set of neighbors of $v\in V(G)$ respectively. A vertex v is said to be a pendant vertex of G if d(v) = 1. Denote by P_n and S_n the path and star of order n respectively. For $u,v\in V(G)$, we use G – v to denote the graph resulting from G by deleting the vertex v and its incident edges. We define G – uv to be the graph obtained from G by deleting the edge $uv\in E(G)$, and G + uv to be the graph obtained from G by adding an edge uv between two non-adjacent vertices u and v of G. The distance d(u, v) between any two vertices u and v in a graph G is the length of a shortest path connecting them. The eccentricity of a vertex $v\in V(G)$ is $e(v)=e_G(v)=\max \{d(u,v):u\in V(G)\}$. The radius of G is $r(G)=\min \{e(v):v\in V(G)\}$. The diameter of G is $d(G)=\max \{e(v):v\in V(G)\}$.

Topological indices play an important role in mathematical chemistry, particularly in studies of quantitative structure property and quantitative structure activity relationships, see e.g. [2, 4, 8]. For a graph G, the harmonic index H(G) is defined in [3] as $$H(G)=\sum _{uv\in E(G)}\frac{2}{d(u)+d(v)},$$where uv is an edge of G, d(u) is the degree of the vertex u in G. Mathematical properties of this index have been studied, see recent review [1], recent paper [6], and the references cited therein.

In 2019, Sowaity et al. [9] consider a closely related variant of the harmonic index, named the eccentric harmonic index which is defined as $$H_e(G)=\sum _{uv\in E(G)}\frac{2}{e(u)+e(v)},$$where uv is an edge of G, e(u) is the eccentric harmonic index of the vertex u in G. Characterizing such graphs with the maximum and minimum the eccentric harmonic index is an interesting work. This motivates our research on the eccentric harmonic index of trees. Recently the authors [7] determined the maximum and minmum eccentric harmonic index of unicyclic graphs with given matching number, and characterize corresponding extremal graphs.

In this paper, we intend to consider the maximum and minmum eccentric harmonic index of trees in terms of graph parameters such as pendant number and matching number, and characterize corresponding extremal graphs.

2 The eccentric harmonic index of trees with given pendant number

Let n, $p\ge 2,k\ge 1$ and $l\ge 1$ be positive integers. Let $\mathcal{T}(n)$ be a family of trees of order n. Let $\mathcal{T}(n,p)$ be a family of trees of order n with p pendant vertices. Let ${\mathcal{T}}_1(n,p)$ be a family of trees order n, which are obtained from two stars S_k and S_l by connecting a path of length $n-p-1$ between their central vertices, where $k+l=p+2$. Particularly, $S_n\in {\mathcal{T}}_1(n,n-1)$. Let S(n, p) be a tree of order n, which is obtained from a central vertex by attaching p pendant paths with a length difference of at most one. Obviously, ${\mathcal{T}}_1(n,p)\subset \mathcal{T}(n,p),S(n,p)\in \mathcal{T}(n,p)$. Note that all the trees in ${\mathcal{T}}_1(n,p)$ have the same eccentric harmonic index, denoted by f(n, p). Let h(n, p) be the eccentric harmonic index of S(n, p).

For a graph G, a path P is called by a pendant path in G if P has exactly one end of degree one. Let $P=v_1{v}_2\dots v_s$ be a pendant path and u be a vertex which is different from V(P) in G. Let $G^{\prime }=G-v_1{v}_2+v_{2}u$. We say $G^{\prime }$ is obtained from G by moving the pendant path P to the vertex u.

Lemma 2.1.

Let $T\in \mathcal{T}(n,p)$. Then $d(T)\le n-p+1$.

Proof.

Suppose contradictory that $d(T)\ge n-p+2$. T contains a longest path $P_{n-p+3}$ of length at least $n-p+2$ and at most $n-(n-p+3)+2=p-1$ pendant vertices, a contradiction. Thus $d(T)\le n-p+1$. □

The upper bound is tight in Lemma 2.1. Let $T\in \mathcal{T}(n,p)$ be a tree obtained from a path $P_{n-p+2}$ of length $n-p+1$ attaching p – 2 new pendant vertices to at least one non-pendant vertices of $P_{n-p+2}$. Then $d(T)=n-p+1$.

Theorem 2.2.

[9] Let T be a tree with n vertices. Then $$H_e(T)\le \frac{2}{3}(n-1)$$with equality if and only if $T=S_n$.

Theorem 2.2 implies that S_n is the unique tree with the maximum eccentric harmonic index in $\mathcal{T}(n)$. We will determine the unique tree with the maximum eccentric harmonic index in $\mathcal{T}(n,p)$, where $2\le p\le n-2$.

Theorem 2.3.

Let $T\in \mathcal{T}(n,p)$, where $2\le p\le n-2$. Then $$H_e(T)\le h(n,p)$$with equality if and only if $T=S(n,p)$.

Proof.

Let $P=v_1{v}_2\dots v_{d(T)}{v}_{d(T)+1}$ be the longest path in T. Clearly, $v_1,v_{d(T)+1}$ are two of p pendant vertices in T.

Take a vertex $w_0\in \{w:\min e_P(w),w\in V(P)\}$, which is a central vertex of P. If there exists one pendant path in T such that its two ends don’t contain w₀, then move such pendant path to w₀. Denote by T₁ such new tree. Note that $$T_1\in \mathcal{T}(n,p),|E(T)|=|E(T_1)|,V(T)=V(T_1)$$

and $$e_T(w)\ge e_{T_1}(w) \mathit{for} \mathit{any} \mathit{vertex} w\in V(T).$$

Thus $H_e(T)\le H_e(T_1).$

Repeat this procedure at most p – 2 times until all pendant paths contain an non-pendant end w₀. Denote by $T^{\prime }$ the resulting graph. Note that $T^{\prime }\in \mathcal{T}(n,p)$ and $H_e(T)\le H_e(T^{\prime })$. Also note that $T^{\prime }$ is a tree with exactly one vertex w₀ of degree p and n – 1 vertices with degree less than or equal to 2. In order to prove our result it suffices to prove the following claim. □

Claim 2.4. Let $T^{\prime }\in \mathcal{T}(n,p)$ be a tree with exactly one vertex of degree p. Then $$H_e(T^{\prime })\le h(n,p)$$with equality if and only if $T=S(n,p)$.

Proof.

Choose the longest and shortest pendant paths in $T^{\prime }$, move the pendant edge of the longest pendant path to the pendant vertex of the shortest pendant path. Denote by T₂ such new tree. Note that $$T_2\in \mathcal{T}(n,p),|E(T^{\prime })|=|E(T_2)|,V(T^{\prime })=V(T_2)$$

and $$e_{T^{\prime }}(w)\ge e_{T_2}(w) \mathit{for} \mathit{any} \mathit{vertex} w\in V(T).$$

Thus $H_e(T^{\prime })\le H_e(T_2).$

Repeat this procedure at most n – p times until all pendant paths have length t or t + 1, where $t=[\frac{n-1}{p}]$. The resulting graph is S(n, p) and $H_e(T^{\prime })\le H_e(S(n,p))=h(n,p).$ This completes the proof of Theorem 2.3. □

Next we will determine the lower bound of the eccentric harmonic index in $\mathcal{T}(n,p)$.

Theorem 2.5.

Let $T\in \mathcal{T}(n,p)$, where $2\le p\le n-1$ and $n\ge 5$. Then $$H_e(T)\ge f(n,p)$$with equality if and only if $T\in {\mathcal{T}}_1(n,p)$.

Proof.

It is trivial for p = 2. Assume that $p\ge 3$. We proceed by induction on n. For n = 4, $T\in {\mathcal{T}}_1(4,3)$. The result follows. Suppose that $n\ge 5$ and it holds for all trees with n – 1 vertices. Let T be a tree with the minimum eccentric harmonic index in $\mathcal{T}(n,p)$. Let P(T) be a set of pendant vertices of T. Let v be a vertex being adjacent to the pendant vertex u in T. □

Claim 2.6. There always exists $u\in P(T)$ such that $H_e(T)=H_e(T-u)+\frac{2}{e(u)+e(v)}.$

Proof.

Let $N^{*}(T)=\{u:e(u)=d(T),u\in P(T)\}$. If $P(T)-N^{*}(T)=\varnothing$, then since $p\ge 3$, there is at least one vertex $u\in P(T)$ such that $d(T-u)=d(T)$. If $P(T)-N^{*}(T)\ne \varnothing$, then take $u\in P(T)-N^{*}(T)$. In above two cases, we have $$e_T(w)=e_{T-u}(w) \mathit{for} \mathit{any} \mathit{vertex} w\in V(T-u).$$

The result follows. □

Assume that the pendant vertex u in T satisfies Claim 2.6. We consider the two following cases:

Case 1.

d(v) = 2.

T – u possesses p pendant vertices. By the induction hypothesis and by Lemma 2.1 and Claim 2.6, $$\begin{array}{c}{H}_e(T)=H_e(T-u)+\frac{2}{e_T(u)+e_T(v)}\\ \ge H_e(T-u)+\frac{2}{2d(T)-1}\\ \ge f(n-1,p)+\frac{2}{2n-2p+1}\\ >f(n,p).\end{array}$$

The last inequality holds since $f(n,p)-f(n-1,p)<\frac{2}{2n-2p+1}$.

Case 2.

d(v) = 3.

T – u possesses p – 1 pendant vertices. By the induction hypothesis and by Lemma 2.1 and Claim 2.6, $$\begin{array}{c}{H}_e(T)=H_e(T-u)+\frac{2}{e(u)+e(v)}\\ \ge H_e(T-u)+\frac{2}{2d(T)-1}\\ \ge f(n-1,p-1)+\frac{2}{2n-2p+1}\\ =f(n,p)\end{array}$$with equalities if and only if $T\in {\mathcal{T}}_1(n,p)$. This completes the proof of Theorem 2.5.

Note that $$\begin{array}{c}f(n,p)\\ =\{\begin{array}{cc}\frac{2p}{2n-2p+1}+4\sum _{i=\frac{n-p+3}{2}}^{n-p}\frac{1}{2i+1}& n-p\equiv 1(\mathit{mod} 2);\\ \frac{2p}{2n-2p+1}+\frac{2}{n-p+4}+4\sum _{i=\frac{n-p+4}{2}}^{n-p}\frac{1}{2i+1}& n-p\equiv 0(\mathit{mod} 2).\end{array}\end{array}$$

Also note that $\frac{2p}{2n-2p+1},\frac{2}{n-p+4}$ and ${\sum }_{i=\frac{n-p+4}{2}}^{n-p}\frac{1}{2i+1}$ are increasing for p, where $2\le p\le n-1$. Thus $f(n,p)\ge f(n,2)$. For $2\le p\le n-2$, S(n, p) has at most two vertices with the eccentricity 2 and at least n – 2 vertices with the eccentricity greater than or equal to 3. Thus $h(n,p)\le (n-2)·\frac{2}{5}+\frac{1}{2}=h(n,n-2)$ for $2\le p\le n-2$.

By Theorems 2.3 and 2.5, we have

Corollary 2.7.

Let $S_n\ne T\in \mathcal{T}(n)$, where $n\ge 5$. Then $$f(n,2)\le H_e(T)\le h(n,n-2)$$with the first equality if and only if $T=P_n$ and the second equality if and only if $T=S(n,n-2)$.

3 The eccentric harmonic index of trees with given matching number

Let $n\ge 2m\ge 2,k\ge 1$, and $l\ge 1$ be positive integers. A matching M of G is a subset of E(G) such that no two edges in M share a common vertex. The cardinality of a maximum matching of G is commonly known as its matching number. M is called the m-matching of G if M contains exactly m edges of G. A vertex v of G is said to be M-saturated if it is incident with an edge of M. If every vertex of G is M-saturated, then M is a perfect matching.

Let ${\mathcal{T}}^{*}(n,m)$ be a family of trees of order n with an m-matching. Let ${\mathcal{T}}_1^{*}(n,m)$ be a family of trees of order $n>2m$, which are obtained from two stars S_k and S_l by connecting a path of length $2m-2$ between their central vertices, where $k+l=n-2m+3$. Let $S^{*}(n,m)$ be a tree of order n, which is obtained from a star $S_{n-m+1}$ by attaching a pendant edge to each of certain m – 1 non-central vertices of $S_{n-m+1}$. Obviously, ${\mathcal{T}}_1^{*}(n,m)\subset {\mathcal{T}}^{*}(n,m),S^{*}(n,m)\in {\mathcal{T}}^{*}(n,m)$. Note that all the trees in ${\mathcal{T}}_1^{*}(n,m)$ have the same eccentric harmonic index, denoted by q(n, m). Let g(n, m) be the eccentric harmonic index of $S^{*}(n,m)$.

Lemma 3.1

([5]). Let $T\in {\mathcal{T}}^{*}(2m,m)$, where $m\ge 3$. Then T has at least two pendant vertices such that each of them is adjacent to a vertex of degree 2.

Lemma 3.2

([5]). Let $T\in {\mathcal{T}}^{*}(n,m)$, where $n>2m$. Then there is an m-matching M and a pendant vertex v such that v is not M-saturated.

Let e = uv be an edge of a graph G. Contract e = uv into a new vertex $u^{\prime }$ and add a new pendant edge $u^{\prime }{v}^{\prime }$ which $v^{\prime }$ is a new pendant vertex. Denote by the resulting graph $G^{\prime }$. We say that $G^{\prime }$ is obtained from G by separating an edge uv.

Lemma 3.3.

Let e = uv be a cut edge of a connected graph G such that $G-uv=G_1\cup G_2$, where $\left|G_1\right|,\left|G_2\right|\ge 2$. Let $G^{\prime }$ be the graph obtained from G by separating the edge uv. Then $H_e(G)<H_e(G^{\prime })$.

Proof.

Let $G^{\prime }$ be a graph which is obtained from G by contracting e = uv into a new vertex $u^{\prime }$ and adding a new pendant edge $u^{\prime }{v}^{\prime }$ which $v^{\prime }$ is a new pendant vertex. Note that $$|E(G^{\prime })|=|E(G)|,|V(G^{\prime })|=|V(G)|$$and $$\begin{array}{c}{e}_{G^{\prime }}(u^{\prime })+e_{G^{\prime }}(v^{\prime })=e_G(u)+e_G(v),\\ e_{G^{\prime }}(w)\le e_G(w) \mathit{for} w\in V(G^{\prime })\backslash \{u^{\prime },v^{\prime }\},\\ e_{G^{\prime }}(u^{\prime })<\min \{e_G(u),e_G(v)\}.\end{array}$$

Thus $$\sum _{\begin{array}{c}{u}^{\prime }w\in E(G^{\prime })\\ w\in V_G\backslash \{u,v\}\end{array}}^{}\mathrm{}\mathrm{}\frac{2}{e_{G^{\prime }}(u^{\prime })+e_{G^{\prime }}(w)}\text{-}\mathrm{}\mathrm{}\sum _{\begin{array}{c}yw\in E(G),y\in \{u,v\}\\ w\in V_G\backslash \{u,v\}\end{array}}^{}\mathrm{}\mathrm{}\frac{2}{e_G(y)+e_G(w)}>0$$and $$\begin{array}{c}{H}_e(G^{\prime })-H_e(G)\hspace{1em}\\ =\sum _{\begin{array}{c}yw\in E(G^{\prime })\\ y,w\in V_G\backslash \{u,v\}\end{array}}^{}(\frac{2}{e_{G^{\prime }}(y)+e_{G^{\prime }}(w)}-\frac{2}{e_G(y)+e_G(w)})\hspace{1em}\\ +\frac{2}{e_{G^{\prime }}(u^{\prime })+e_{G^{\prime }}(v^{\prime })}+\sum _{\begin{array}{c}{u}^{\prime }w\in E(G^{\prime })\\ w\in V_G\backslash \{u,v\}\end{array}}^{}\frac{2}{e_{G^{\prime }}(u^{\prime })+e_{G^{\prime }}(w)}\hspace{1em}\\ -\frac{2}{e_G(u)+e_G(v)}-\sum _{\begin{array}{c}yw\in E(G),y\in \{u,v\}\\ w\in V_G\backslash \{u,v\}\end{array}}^{}\frac{2}{e_G(y)+e_G(w)}\hspace{1em}\\ >0.\end{array}$$

This completes the proof of Lemma 3.3. □

We now prove the main results of this section.

Theorem 3.4.

Let $T\in {\mathcal{T}}^{*}(n,m)$. Then if $n=2m$, then $$H_e(T)\ge H_e(P_{2m});$$if $n>2m$, then $$H_e(T)\ge q(n,m);$$ with equality if and only if $T\in {\mathcal{T}}_1^{*}(n,m)$.

Proof.

Let $T\in {\mathcal{T}}^{*}(n,m)$. Let $u,v\in V(T)$ such that $e(u)=e(v)=d(T)$. Obviously, u and v are both pendant vertices. If T has at least one pendant vertex w which is different from u and v. Move the pendant edge containing w to the vertex u. Denote by T₁ such new tree. Note that $$|E(T)|=|E(T_1)|,V(T)=V(T_1)$$

and $$e_T(w)\le e_{T_1}(w) \mathit{for} \mathit{any} \mathit{vertex} w\in V(T).$$

Thus $H_e(T)\ge H_e(T_1).$

Repeat this procedure at most $2m-d(T)$ times until the resulting graph $T^{\prime }$ contains the longest path in ${\mathcal{T}}^{*}(n,m).$ Thus $H_e(T)\ge H_e(T^{\prime })$. If $n=2m$, then $T^{\prime }=P_{2m}$. If $n>2m$, then let $P=v_1{v}_2\dots v_{2m+1}$ be the longest path in $T^{\prime }$. Obviously, $V(T^{\prime })\backslash \{v_1,v_2,\dots ,v_{2m+1}\}$ are pendant vertices. Otherwise, there exist a pendant path of length at least 2. We can get a $(m+1)$-matching obtained by taking a pendant edge in such pendant path and m mutually independent edges in P, a contradiction. Similarly, move an pendant edge except $v_{2m}{v}_{2m+1}$ and $v_1{v}_2$ to the vertices v₂ or $v_{2m}$ at a time. Repeat this procedure at most $n-2m-1$ times until the resulting graph $T^{″}$ is a graph in ${\mathcal{T}}_1^{*}(n,m)$. Thus $H_e(T)\ge H_e(T^{″}).$ This completes the proof of Theorem 3.4. □

Theorem 3.5.

Let $T\in {\mathcal{T}}^{*}(2m,m)$. Then $$H_e(T)\le g(2m,m)$$with equality if and only if $T\cong S^{*}(2m,m)$.

Proof.

We proceed by induction on m. If m = 1, 2 then $T=S^{*}(2m,m)=P_{2m}$. If m = 3, then $T=S^{*}(2m,m)$ or $T=P_{2m}$. Note that $H_e(P_{2m})<H_e(S^{*}(2m,m))$. The result follows. Assume that $m\ge 4$.

Let M be a perfect matching in T. If these is a non-pendent edge $xy\in M$, then let T⁰ be obtained from T by separating an edge xy. Thus T⁰ contains a perfect matching $M_1=M\cup \left\{x^{\prime }{y}^{\prime }\right\}\backslash \left\{xy\right\}$. Repeat this procedure until there is no non-pendant edge in the most updated perfect matching. Suppose that the resulting tree and the corresponding perfect matching are $T^{\prime }$ and $M^{\prime }$ respectively. If at least one separating is preformed, then by Lemma 3.3, $$H_e(T)<H_e(T^{\prime }).$$

We can assume that each edge in $M^{\prime }$ is a pendant edge.

By Lemma 3.1, $T^{\prime }$ has a pendant vertex u which is adjacent to a vertex v of degree 2. Let $N(v)=\{w,u\}$. Let $T^{*}=T^{\prime }-u-v$. Then $T^{*}\in {\mathcal{T}}^{*}(2m-2,m-1)$. Let d(w) = d and $N(w)\backslash \left\{v\right\}=\{x_1,x_2,\dots ,x_{d-1}\}$. Recall that $M^{\prime }$ is a perfect matching which contains only pendant edges. Thus $d(w)\ge 3$. Otherwise, $M^{\prime }$ has an non-pendant edge containing an end w, a contradiction. Without loss of generality, we can assume $x_{1}w\in M^{\prime }$ and hence $d(x_1)=1,d(x_i)\ge 2$ for $i=2,3,\dots ,d-1$.

Note that $$e_{T^{\prime }}(y)\ge e_{T^{*}}(y) \mathit{for} \mathit{any} \mathit{vertex} y\in V(T^{*}).$$

Thus (3.1) $$\sum _{\begin{array}{c}{w}_1{w}_2\in E(T^{*})\\ w_1{w}_2\in V_{T^{*}}\end{array}}^{}(\frac{2}{e_{T^{\prime }}(w_1)+e_{T^{\prime }}(w_2)}-\frac{2}{e_{T^{*}}(w_1)+e_{T^{*}}(w_2)})\le 0$$(3.1)

Also note that $$\begin{array}{c}{H}_e(T^{\prime })-H_e(T^{*})\\ =\sum _{\begin{array}{c}{w}_1{w}_2\in E(T^{*})\\ w_1{w}_2\in V_{T^{*}}\end{array}}^{}(\frac{2}{e_{T^{\prime }}(w_1)+e_{T^{\prime }}(w_2)}-\frac{2}{e_{T^{*}}(w_1)+e_{T^{*}}(w_2)})\\ +\frac{2}{e_{T^{\prime }}(u)+e_{T^{\prime }}(v)}+\frac{2}{e_{T^{\prime }}(w)+e_{T^{\prime }}(v)}\\ \le \frac{2}{e_{T^{\prime }}(u)+e_{T^{\prime }}(v)}+\frac{2}{e_{T^{\prime }}(w)+e_{T^{\prime }}(v)}\\ \le \frac{2}{4r(T^{\prime })-1}+\frac{2}{4r(T^{\prime })-3}.\end{array}$$

Recall that $m\ge 4$. By Lemma 3.1, $r(T^{\prime })\ge 2$. Thus (3.2) $$\begin{array}{c}{H}_e(T^{\prime })-H_e(T^{*})\le \frac{2}{4r(T^{\prime })-1}+\frac{2}{4r(T^{\prime })-3}\\ \le \frac{24}{35}\end{array}$$(3.2)

with equality if and only if $r(T^{\prime })=2$. By the induction hypothesis, we have $$\begin{array}{c}{H}_e(T^{\prime })\le H_e(T^{*})+\frac{24}{35}\\ \le g(2m-2,m-1)+\frac{24}{35}\\ =g(2m,m)\end{array}$$with equalities (3.1) and (3.2) holding if and only if $T\cong S^{*}(2m,m)$. This completes the proof of Theorem 3.5. □

Let $T\in {\mathcal{T}}^{*}(n,2)$, where n > 4. Then it is easy to verify that $H_e(T)\le g(n,2)$ with equality if and only if $T\in {\mathcal{T}}_1^{*}(n,2)$ and d(T) = 3. Clearly, $S^{*}(n,2)$ is only one of the extremal graphs with the maximum eccentric harmonic index in ${\mathcal{T}}^{*}(n,2)$.

Theorem 3.6.

Let $T\in {\mathcal{T}}^{*}(n,m)$ and $n\ge 2m\ge 6$. Then $$H_e(T)\le g(n,m)$$with equality if and only if $T\cong S^{*}(n,m)$.

Proof.

We proceed by induction on n. If $n=2m,$ then by Theorem 3.5, the result follows. Suppose that $n>2m$ and it holds for all trees in ${\mathcal{T}}^{*}(n-1,m)$.

By Lemma 3.2, T has an m-matching M and a pendant vertex v which is not M-saturated. Let $u\in V(T)$ with $N(u)=\{v,v_1,v_2,\dots ,v_{d-1}\}$ and d(u) = d. Let $T^{\prime }=T-v$. Then $T^{\prime }\in {\mathcal{T}}^{*}(n-1,m)$.

Note that $$e_T(y)\ge e_{T^{\prime }}(y) \mathit{for} \mathit{any} \mathit{vertex} y\in V(T^{\prime }).$$

Thus (3.3) $$\sum _{\begin{array}{c}{w}_1{w}_2\in E(T^{\prime })\\ w_1{w}_2\in V_{T^{\prime }}\end{array}}^{}(\frac{2}{e_T(w_1)+e_T(w_2)}-\frac{2}{e_{T^{\prime }}(w_1)+e_{T^{\prime }}(w_2)})\le 0$$(3.3)

Also note that $$\begin{array}{c}{H}_e(T)-H_e(T^{\prime })\\ =\sum _{\begin{array}{c}{w}_1{w}_2\in E(T^{\prime })\\ w_1{w}_2\in V_{T^{\prime }}\end{array}}^{}(\frac{2}{e_T(w_1)+e_T(w_2)}-\frac{2}{e_{T^{\prime }}(w_1)+e_{T^{\prime }}(w_2)})\\ +\frac{2}{e_T(u)+e_T(v)}\\ \le \frac{2}{e_T(u)+e_T(v)}\\ \le \frac{2}{2r(T)+1}.\end{array}$$

Recall that $m\ge 3$. Thus $r(T)\ge 2$ and (3.4) $$H_e(T)-H_e(T^{\prime })\le \frac{2}{5}.$$(3.4)

with equality if and only if r(T) = 2. By the induction hypothesis, we have $$\begin{array}{c}{H}_e(T)\le H_e(T^{\prime })+\frac{2}{5}\\ \le g(n-1,m)+\frac{2}{5}\\ =g(n,m)\end{array}$$with equalities (3.3) and (3.4) holding if and only if $T\cong S^{*}(n,m)$. This completes the proof of Theorem 3.6. □

Acknowledgments

The authors sincerely appreciate the reviewers for all the valuable comments and suggestions, which helped us to improve the quality of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the innovation project of Guangdong Education Department (No. 2020KTSCX078), the Natural Science Foundation of Guangdong Province (Nos. 2022A1515011551, 2022A1515011971), the 13th Five-Year Plan subject of Guangdong Province (No. GD20XGL25), and projects of Hanshan Normal University (Nos. QN202024, QN202025, 2020220).