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On the $A_α$-characteristic polynomials of some join graphs

Mainak Basunia, Pratima Panigrahi

Abstract

For $α\in [0,1]$, the $A_α$-matrix of graph $G$ is $A_α(G) = αD(G) + (1- α) A(G)$, where $A(G)$ and $D(G)$ are adjacency and degree diagonal matrix of $G$, respectively. Let $G_1 \dot{\vee}_Q G_2$, $G_1 \underline{\vee}_Q G_2$, $G_1 \dot{\vee}_T G_2$ and $G_1 \underline{\vee}_T G_2$ represent the $Q$-vertex join, $Q$-edge join, $T$-vertex join and $T$-edge join respectively, obtained from the graphs $G_1$ and $G_2$. This article determines the $A_α$-characteristic polynomials of the graphs $G_1 \dot{\vee}_Q G_2$, $G_1 \underline{\vee}_Q G_2$, $G_1 \dot{\vee}_T G_2$ and $G_1 \underline{\vee}_T G_2$ where $G_1$ is regular and $G_2$ can be any graph. The $A_α$-spectra of these graphs are computed in terms of those of $G_1$ and $G_2$ for both $G_1$ and $G_2$ are regular, as well as for $G_1$ is regular and $G_2$ is $K_{a,b}$, a complete bipartite graph. As an application, we construct an infinite number of pairs of graphs that have the same $A_α$-spectrum.

On the $A_α$-characteristic polynomials of some join graphs

Abstract

For , the -matrix of graph is , where and are adjacency and degree diagonal matrix of , respectively. Let , , and represent the -vertex join, -edge join, -vertex join and -edge join respectively, obtained from the graphs and . This article determines the -characteristic polynomials of the graphs , , and where is regular and can be any graph. The -spectra of these graphs are computed in terms of those of and for both and are regular, as well as for is regular and is , a complete bipartite graph. As an application, we construct an infinite number of pairs of graphs that have the same -spectrum.
Paper Structure (7 sections, 24 theorems, 69 equations, 5 figures)

This paper contains 7 sections, 24 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $A_{\alpha}$-characteristic polynomial of $Q$-vertex join
  4. $A_{\alpha}$-characteristic polynomial of $Q$-edge join
  5. $A_{\alpha}$-characteristic polynomial of $T$-vertex join
  6. $A_{\alpha}$-characteristic polynomial of $T$-edge join
  7. Concluding remarks

Key Result

Lemma 2.1

spec_of_subvertexjoin_subedgejoin_by_Liu__Zhang For a real number $b$ and a real matrix $M$ of size $n\times n$, we have where $\mathop{\mathrm{adj}}\nolimits(M)$ denotes the adjugate of $M$.

Figures (5)

  • Figure 1: $Q$-vertex join, $Q$-edge join, $T$-vertex join and $T$-edge join of $P_4$ and $P_3$
  • Figure 2: The graph $H$
  • Figure 3: The graph $\tilde{H}$
  • Figure 4: Two $L_S$-cospectral graphs $G_1$ and $G_2$
  • Figure 5: Two new $L_S$-cospectral graphs $G_1\dot{\vee}_Q P_2$ and $G_2\dot{\vee}_Q P_2$

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4: Schur complement formula
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Theorem 3.1
  • proof
  • ...and 26 more