On Hosoya's dormants and sprouts

TL;DR

This paper provides explicit formulas and conditions for constructing cospectral graphs through multiple coalescences, expanding understanding of their properties and prevalence.

Contribution

It introduces an explicit characteristic polynomial formula for multiple coalescences and characterizes their cospectrality, addressing questions raised by Hosoya.

Findings

Derived a necessary and sufficient condition for cospectrality of multiple coalescences.

Enumerated cospectral multiple coalescences for specific graph families.

Proved the infinitude of cospectral coalescences with path graphs as underlying graphs.

Abstract

In a recent series of papers, Hosoya drew the attention to a particular aspect of constructing cospectral graphs by using coalescences: that cospectral graphs can be constructed by attaching multiple copies of a rooted graph in different ways to subsets of vertices of an underlying graph. Our principal focus is to address the expectations and questions raised in Hosoya's papers with regards to this construction. We give an explicit formula for the characteristic polynomial of such multiple coalescences, from which we obtain a necessary and sufficient condition for their cospectrality. We enumerate such cospectral multiple coalescences for a few families of underlying graphs, and show the infinitude of cospectral multiple coalescences having paths as underlying graphs, which were deemed rare by Hosoya.