Corrigendum to ‘Complex adjacency matrix and energy of digraphs’

ABSTRACT

In Section 4 of the original paper ‘Complex adjacency matrix and energy of digraphs’, we incorrectly asserted that the iota energy of digraphs increases with respect to the quasi-order relation defined over $D_{n,h}$ if $h\not\equiv 0\left(mod4\right)$. In this corrigendum, we point out errors and correct them where possible.

This article refers to:

Complex adjacency matrix and energy of digraphs

1. Increasing property of iota energy

In Section 4 of the original paper [1], we incorrectly calculated the formulas (4.14) and (4.15) of energy and iota energy, respectively, for the polynomial of the form $\phi \left(x\right)=x^h-(a+\dot{\iota }b)$. The correct formulas for energy and iota energy of the polynomial $\phi \left(x\right)$ are given below: $E\left(\phi \right(x\left)\right)=\left\{\begin{array}{ll}{r}^{1/h}\left(\cos \frac{\theta }{h}E\left(C_h\right)+2\left|\sin \frac{\theta }{h}\right|\right)& \text{if }h\equiv 0\left(mod4\right)\\ r^{1/h}\cos \frac{\theta }{h}E\left(C_h\right)& \text{if }h\equiv 2\left(mod4\right)\\ r^{1/h}\left(\cos \frac{\theta }{h}E\left(C_h\right)-2\sin \left(\frac{\theta }{h}+\frac{\pi }{2h}\right)\right)& \text{if }h\equiv 1\left(mod2\right)\text{and}\\ & -\pi <\theta <-\frac{\pi }{2}\\ r^{1/h}\cos \frac{\theta }{h}E\left(C_h\right)& \text{if }h\equiv 1\left(mod2\right)\text{ and }\\ & -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}\\ r^{1/h}\left(\cos \frac{\theta }{h}E\left(C_h\right)+2\sin \left(\frac{\theta }{h}-\frac{\pi }{2h}\right)\right)& \text{if }h\equiv 1\left(mod2\right)\text{ and }\frac{\pi }{2}\le \theta \le \pi \end{array}\right.$ and $E_c\left(\phi \right(x\left)\right)=\left\{\begin{array}{ll}{r}^{1/h}\left(\cos \frac{\theta }{h}{E}_c\left(C_h\right)+2\left|\sin \frac{\theta }{h}\right|\right)& \text{if }h\equiv 0\left(mod2\right)\\ r^{1/h}\left(\cos \frac{\theta }{h}{E}_c\left(C_h\right)+\left|\sin \frac{\theta }{h}\right|\right)& \text{if }h\equiv 1\left(mod2\right),\end{array}\right.$ where θ is the principal argument of $a+\dot{\iota }b$ and $r=\sqrt{a^2+b^2}$.

The statement of Theorem 4.2 [1] is flawed as it was proved using the formulas (4.14) and (4.15). In the next theorem, we give correct statement of Theorem 4.2 [1]. The proof is similar.

Theorem 1.1

Let $D\in {\mathcal{D}}_{n,h}$ be a digraph. Then there are functions $f_1\left(h\right),f_2\left(h\right)$ and $f_3\left(h\right)$ such that $\begin{array}{rl}E\left(D\right)& =\left\{\begin{array}{ll}{f}_1\left(h\right)E\left(C_h\right)+2f_2\left(h\right)& \text{if }h\equiv 0\left(mod4\right)\\ f_1\left(h\right)E\left(C_h\right)& \text{if }h\equiv 2\left(mod4\right)\\ f_1\left(h\right)E\left(C_h\right)+f_3\left(h\right)& \text{if }h\equiv 1\left(mod2\right),\end{array}\right.\\ E_c\left(D\right)& =\left\{\begin{array}{ll}{f}_1\left(h\right)E_c\left(C_h\right)+2f_2\left(h\right)& \text{if }h\equiv 0\left(mod2\right)\\ f_1\left(h\right)E_c\left(C_h\right)+f_2\left(h\right)& \text{if }h\equiv 1\left(mod2\right).\end{array}\right.\end{array}$

The following expression is proved correctly in Theorem 4.5 [1]: $E_c\left(D\right)=\frac{1}{\pi }p.v{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{x^2}\log \left(1+\sum _{k=1}^{\lfloor n/h\rfloor }c(D,kh)x^{kh}\right)\mathrm{d}x.$ Based on the above expression, we erroneously stated that the iota energy increases with respect to the quasi-order relation ≼ defined over $D_{n,h}$ if $h\equiv 1\left(mod2\right)$. The digraphs shown in Figure 1 disprove Theorem 4.5 [1]. That is, the iota energy in general does not possess the increasing property over the set ${\mathcal{D}}_{n,h}$ when $h\equiv 1\left(mod2\right)$.

Figure 1. $D_1,D_2\in {\mathcal{D}}_{26,5}$.

Theorem 4.6 [1] was proved using Theorem 4.2 [1] and thus the assertion of Theorem 4.6 [1] is not correct. The digraphs shown in Figure 2 disprove Theorem 4.6 [1]. That is, the iota energy in general may not possess the increasing property over the set $D_{n,h}$ when $h\equiv 2\left(mod4\right)$.

Figure 2. $D_3,D_4\in {\mathcal{D}}_{22,6}$.

Finally, Theorem 4.7 [1] was a consequence of Theorems 4.2 and 4.5 [1] and thus may not be correct. However, we failed to find a counter example to disprove the Theorem 4.7 [1].