On the characteristic polynomial of the $A_\alpha$-matrix for some operations of graphs

TL;DR

This paper derives the characteristic polynomial of the $A_eta$-matrix for graphs formed through various operations, including coalescing and line graph construction, extending spectral graph theory results.

Contribution

It provides explicit formulas for the $A_eta$-characteristic polynomial for graphs obtained by specific operations, such as coalescing and line graph formation, for regular and semi-regular bipartite graphs.

Findings

Derived the $A_eta$-characteristic polynomial for coalesced graphs.

Obtained the $A_eta$-characteristic polynomial for line graphs of semi-regular bipartite graphs.

Presented the $A_eta$-characteristic polynomial for graphs from certain operations on regular graphs.

Abstract

Let G be a graph of order $n$ with adjacency matrix $A(G)$ and diagonal matrix of degree $D(G)$. For every $\alpha \in [0,1]$, Nikiforov \cite{VN17} defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$. In this paper we present the $A_{\alpha}(G)$-characteristic polynomial when $G$ is obtained by coalescing two graphs, and if $G$ is a semi-regular bipartite graph we obtain the $A_{\alpha}$-characteristic polynomial of the line graph associated to $G$. Moreover, if $G$ is a regular graph we exhibit the $A_{\alpha}$-characteristic polynomial for the graphs obtained from some operations.