Some Graphs Determined by their Signless Laplacian (Distance) Spectra

Main Article Content

Chandrashekar Adiga
Kinkar Das
B. R. Rakshith

Abstract

In literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtained from the complete graph $K_{n}$ by removing all the edges of a complete bipartite subgraph $K_{l,m}$ is studied. It is shown that the graph $K_{n}\backslash K_{1,m}$ with $m\ge4$ is determined by its signless Laplacian spectrum, and it is proved that the graph $K_{n}\backslash K_{l,m}$ is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph $K_{n-2\alpha}\vee \alpha K_{2}$ was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em Linear Algebra Appl.}, 436:3634--3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for $n-2\alpha=1 ~\text{or}~ 2$. Here, this problem is completely solved for all positive integer $n-2\alpha$. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.

Article Details

Section
Article