Construction of non-regular $A_\alpha$-cospectral graphs from some join of graphs

TL;DR

This paper computes the $A_\alpha$-spectra of various graph joins and constructs non-regular, non-isomorphic graphs that are $A_\alpha$-cospectral, advancing understanding of spectral graph theory.

Contribution

It introduces methods to compute $A_\alpha$-spectra for specific graph joins and constructs new non-regular, non-isomorphic $A_\alpha$-cospectral graphs.

Findings

Computed $A_\alpha$-characteristic polynomials for multiple graph joins.

Derived $A_\alpha$-spectra for regular graphs in these joins.

Constructed examples of non-regular, non-isomorphic $A_\alpha$-cospectral graphs.

Abstract

Cospectral graphs are a fascinating concept in graph theory, where two non-isomorphic graphs possess identical sets of eigenvalues. In this paper, we compute the $A_\alpha$-characteristic polynomial of neighbour and non-neighbour splitting join, neighbour and non-neighbour shadow join, central vertex and edge join and duplicate join of two graphs. In addition, when $\graphene_1$ and $\graphene_2$ are regular, we compute the $A_\alpha$-spectrum of these graphs. As an application, we construct non-regular, non-isomorphic graphs that are $A_\alpha$-cospectral.