Complements of coalescing sets

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...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Authors: Butler, Steve, D'Avanzo, Elena, Heikkinen, Rachel, Jeffries, Joel, Kruczek, Alyssa, Harper Niergarth
Format: Article
Language:English
Subjects:
Polynomials
Online Access:Get full text
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Summary: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 \(G\).
ISSN:2331-8422