The strong spectral property for some families of unicyclic graphs

TL;DR

The paper addresses the problem of classifying graphs whose symmetric off-diagonal patterns force the Strong Spectral Property (SSP) for all realizations, with implications for the inverse eigenvalue problem on graphs (IEP-$G$). It develops and utilizes the SSP verification matrix framework to analyze ${\mathcal{S}}(G)$ and related sets, enabling a structural, girth-based study of unicyclic graphs. The main contributions include a complete characterization of unicyclic graphs of girth $3$ in ${\mathcal{G}}^{\text{SSP}}$ and the demonstration that all tadpole graphs with girth up to $5$ belong to ${\mathcal{G}}^{\text{SSP}}$, while girth-$6$ tadpoles are not, along with detailed results for girth $4$ and $5$ and several explicit non-SSP examples to delimit the boundary. These results advance the sparse-graph classification of SSP graphs and clarify how cycle-bridge structures influence SSP, informing future investigations into higher girth cases and SSP-preserving operations.

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.

Figures (3)

• Figure 1: Graph $Z_3$ together with its strong powers $Z_3^{(r)}$, $r\leq 3$.
• Figure 2: Graph $G$ and its supergraphs $G_{\ell}$, $\ell\in [8]$, illustrating the steps in the proof of Theorem \ref{['thm:unicyclic-girth-3-graphs']}. The blue edges in $E(G_{\ell})$, $\ell\in [8]$, are the edges in $E(G_{\ell})\setminus E(G_{\ell-1})$, and the grey edges are the ones already in $E(G_{\ell-1})$.
• Figure 3: Five examples of unicyclic graphs of girth four not in ${\mathcal{G}}^{\text{SSP}}$, see Examples \ref{['ex:4cycle+4paths']}, \ref{['ex:4cycle+3paths']} and \ref{['ex:even-cycle+2paths']}.

Theorems & Definitions (24)

• Definition 2.1
• Example 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 3.1
• proof
• Theorem 3.2