$A_{\alpha}$-Spectra of $Q$- and $T$-Join Graphs with Applications to Cospectral Constructions

TL;DR

This paper investigates the $A_{\alpha}$-spectra of graphs formed through specific join operations, providing explicit formulas and enabling efficient spectral analysis of complex graphs for applications like cospectral graph construction.

Contribution

It derives explicit $A_{\alpha}$-characteristic polynomials and spectra for graphs created via $Q$- and $T$-join operations, especially when factor graphs are regular or complete bipartite.

Findings

Explicit formulas for $A_{\alpha}$-characteristic polynomials of join graphs.

Complete $A_{\alpha}$-spectra expressed in terms of factor graphs' spectra.

Construction of infinitely many $A_{\alpha}$-cospectral non-isomorphic graphs.

Abstract

For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of a graph $G$ is defined by $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal degree matrix of $G$, respectively. In this paper, we study the $A_{\alpha}$-characteristic polynomials and $A_{\alpha}$-spectra of graphs obtained via four recently introduced join operations, namely the $Q$-vertex join, $Q$-edge join, $T$-vertex join, and $T$-edge join, applied to two graphs $G_1$ and $G_2$. We derive explicit expressions for the $A_{\alpha}$-characteristic polynomials of these constructions when the first factor graph is regular. Furthermore, we determine the complete $A_{\alpha}$-spectra of these graphs in terms of the $A_{\alpha}$-spectra of the factor graphs, particularly when the second factor graph is regular or complete bipartite. The significance of these results lies in the fact that they enable efficient computation of the $A_{\alpha}$-spectra of large complex graphs arising from these joins, directly from the $A_{\alpha}$-spectra of the smaller constituent graphs, without explicitly constructing and handling the complex $A_{\alpha}$-matrices of those large graphs. Finally, as an application, we demonstrate how to construct infinitely many families of non-isomorphic graphs that are $A_{\alpha}$-cospectral.