Determining some graph joins by the signless Laplacian spectrum
Jiachang Ye, Jianguo Qian, Zoran Stanić
TL;DR
The paper investigates which graphs are determined by the signless Laplacian spectrum (DQS) within a natural family formed by taking a join with a single vertex: $G\cong K_1\vee (C_{l_1}\cup\cdots\cup C_{l_t}\cup sK_1)$. It derives explicit $Q$-spectra for these joins, analyzes eigenvalue multiplicities and degree constraints, and proves that such graphs are DQS for $n\ge 22$ exactly when $s=0$ or $s\ge1$ with all $l_i\neq 3$; if a final cycle has length $3$, a Q-cospectral partner exists by replacing that $C_3$ with a $K_{1,3}$-related structure. The results extend prior work on $Q$-cospectral joins and contribute to the broader program of characterizing graphs by their signless Laplacian spectra. The approach combines explicit spectrum computations with interlacing-type arguments and careful degree-structure analysis in joins.
Abstract
A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let $C_l$, $P_l$, $K_l$ and $K_{s,l-s}$ be the cycle, the path, the complete graph and the complete bipartite graph with $l$ vertices, respectively. We prove that $$G\cong K_1\vee (C_{l_1}\cup C_{l_2}\cup\cdots \cup C_{l_t}\cup sK_1),$$ with $s\ge 0, t\ge 1, n\geq 22$, is determined by the signless Laplacian spectrum if and only if either $s=0$ or $s\ge 1$ and $l_i\ne 3$ holds for all $1\leq i\leq t$, where $n$ is the order of $G$, and $\cup$ and $\vee$ stand for the disjoint union and the join of two graphs, respectively. Moreover, for $s\ge 1$ and $l_t=3$, $K_1\vee (K_{1,3}\cup C_{l_1}\cup C_{l_2}\cup\cdots \cup C_{l_{t-1}}\cup (s-1)K_1)$ is fixed as a graph sharing the signless Laplacian spectrum with $G$. This contribution extends some recently published results.