Signless Laplacian characterization of cones over disjoint unions of cycles, edges and isolated vertices

Abstract

Two graphs are said to be $Q$-cospectral if they share the same signless Laplacian spectrum. A simple graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if there exists no other non-isomorphic simple graph with the same signless Laplacian spectrum. In this paper, we establish the following results: (1) Let$G\cong K_1\vee (C_k\cup q{K}_2\cup s{K}_1),$ with $q,s\ge 1$, $k\ge 4$, and at least $21$ vertices. If $k$ is odd, then $G$ is DQS. Moreover, if $k$ is even and $F$ is $Q$-cospectral with $G$, then $$F\cong G\text{or}F\cong K_1\vee (C_4\cup P_{k-3}\cup P_3\cup (q-2)K_2\cup s{K}_1).$$ (2) Let $G\cong K_1\vee (C_{k_1}\cup C_{k_2}\cup \cdots \cup C_{k_t}\cup q{K}_2\cup s{K}_1)$ with $t\ge 2$, $q,s\ge 1$, $k_i\ge 4$ and at least $33$ vertices. If each $k_i$ is odd, then $G$ is DQS. (3) The graph $K_1\vee (C_3\cup C_{k_1}\cup C_{k_2}\cup \cdots \cup C_{k_{t-1}}\cup q{K}_2\cup s{K}_1),$ with $t,q,s\ge 1$ and $k_i\ge 3$, is not DQS. Moreover, it is $Q$-cospectral with $K_1\vee (K_{1,3}\cup C_{k_1}\cup C_{k_2}\cup \cdots \cup C_{k_{t-1}}\cup q{K}_2\cup (s-1)K_1).$ Here $P_n$, $C_n$, $K_n$ and $K_{n-r,r}$ denote the path, the cycle, the complete graph and the complete bipartite graph on $n$ vertices, while $\cup$ and $\vee$ represent the disjoint union and the join of two graphs, respectively. Furthermore, the signless Laplacian spectrum of the graphs under consideration is computed explicitly.

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