Laplacian Spectral Determination of Path-Friendship Graphs

TL;DR

This paper proves that path-friendship graphs are uniquely determined by their Laplacian spectrum, extending known results about friendship graphs and starlike trees, and contributing to the spectral characterization of graph families.

Contribution

It establishes that path-friendship graphs are determined by their Laplacian spectrum, a new class shown to have this spectral property.

Findings

Path-friendship graphs are Laplacian spectral determined.

Extension of spectral determination from friendship graphs to path-friendship graphs.

Supports the conjecture that many graph families are DLS.

Abstract

A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to $G$. van Dam and Haemers (2003) conjectured that almost all graphs have this property, but that is known to be the case only for a very few families. In some recent papers it is proved that the friendship graphs and starlike trees are DLS. If a friendship graph and a starlike tree are joined by merging their vertices of degree greater than 2, then the resulting graph is called a path-friendship graph. In this paper, it is proved that the path-friendship graphs are also DLS.