The strong spectral property for some families of unicyclic graphs

TL;DR

This paper characterizes unicyclic graphs with girth three and certain tadpole graphs with girth up to five that possess the Strong Spectral Property, crucial for understanding spectra of symmetric matrices patterned by these graphs.

Contribution

It provides a complete characterization of unicyclic graphs of girth three and identifies which tadpole graphs up to girth five have the Strong Spectral Property.

Findings

Unicyclic graphs of girth three are in ${ m SSP}$.

All tadpole graphs with girth at most five are in ${ m SSP}$.

Girth six tadpole graphs do not necessarily have the Strong Spectral Property.

Abstract

To find all the possible spectra of all real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of a given graph $G$, the Strong Spectral Property turned out to be of crucial importance. In particular, we investigate the set ${\mathcal G}^{\text{SSP}}$ of all simple graphs $G$ with the property that each symmetric matrix of the pattern of $G$ has the Strong Spectral Property. In this paper, we completely characterize unicyclic graphs of girth three in ${\mathcal G}^{\text{SSP}}$. We prove that any tadpole graph of girth at most five is in ${\mathcal G}^{\text{SSP}}$ and we show that the same is not valid for girth six tadpole graphs.