Which $L$-cospectral graphs have same degree sequences

TL;DR

This paper investigates conditions under which Laplacian cospectral graphs with the same degree sequences can be uniquely identified, providing new spectral characterizations for certain graph classes.

Contribution

It establishes spectral conditions that guarantee degree sequence uniqueness for Laplacian cospectral graphs, extending known results to broader graph families.

Findings

Graphs with specific Laplacian eigenvalue bounds have identical degree sequences if they are cospectral.

Multi-fan graphs are uniquely determined by their Laplacian spectrum.

Certain composite graphs with odd cycles are also uniquely identified by their Laplacian spectrum.

Abstract

Let $\lambda_{i}(G)$ be the $i$-th largest Laplacian eigenvalues of graph $G$, where $1\le i\le |V(G)|$. Liu, Yuan, You and Chen [Discrete Math., 341 (2018) 2969--2976] raised the problem for ``Which cospectral graphs have same degree sequences". In this paper, let $W_3$ and $W_5$ be the two graphs as shown in Fig. 2 and let $G$ be a connected graph with $n\ge 18$ vertices. We shall show that: $(1)$ If $\lambda_{2}(G)<5<n-1<\lambda_{1}(G)$, $\lambda_{1}(G) \notin \{\lambda_{1}(W_3),\lambda_{1}(W_5)\}$ and $H$ is Laplacian cospectral with $G$, then $H$ must have the same degree sequence with $G$; $(2)$ If $\lambda_2(G)\le 4.7<n-2< \lambda_1(G)$, and $H$ is Laplacian cospectral with $G$, then $H$ must have the same degree sequence with $G$. The former result easily leads to the unique theorem result of [Discrete Math., 308 (2008) 4267--4271], that is: Every multi-fan graph $K_1\vee (P_{l_1}\cup P_{l_1}\cup\cdots \cup P_{l_t})$ is determined by the Laplacian spectrum. Moreover, it can also deduce a new conclusion: $K_1\vee (P_{l_1}\cup P_{l_1}\cup\cdots \cup P_{l_t}\cup C_{s_1}\cup C_{s_2}\cup\cdots \cup C_{s_k})$ $(t\ge 1, k\ge 1)$ is determined by the Laplacian spectrum if the graph order $n\ge 18$ and each $s_i$ $(i=1,2,\ldots, k)$ is odd.