Complements of coalescing sets

TL;DR

This paper derives a formula for the characteristic polynomial of matrices formed by coalescing graphs and establishes cospectrality conditions for such graph constructions, advancing spectral graph theory understanding.

Contribution

It introduces a formula for the characteristic polynomial of coalesced graphs and proves cospectrality preservation under certain graph modifications.

Findings

Derived a general formula for the characteristic polynomial of coalesced graphs.

Proved that cospectral pairs remain cospectral after specific vertex removals.

Established conditions under which coalescent pairs are cospectral for all rooted graphs.

Abstract

We consider matrices of the form $qD+A$, with $D$ being the diagonal matrix of degrees, $A$ being the adjacency matrix, and $q$ a fixed value. Given a graph $H$ and $B\subseteq V(G)$, which we call a coalescent pair $(H,B)$, we derive a formula for the characteristic polynomial where a copy of same rooted graph $G$ is attached by the root to \emph{each} vertex of $B$. Moreover, we establish if $(H_1,B_1)$ and $(H_2,B_2)$ are two coalescent pairs which are cospectral for any possible rooted graph $G$, then $(H_1,V(H_1)\setminus B_1)$ and $(H_2,V(H_2)\setminus B_2)$ will also always be cospectral for any possible rooted graph $G$.