15 research outputs found
Connected graphs cospectral with a Friendship graph
Let be any positive integer, the friendship graph consist of
edge-disjoint triangles that all of them meeting in one vertex. A graph is
called cospectral with a graph if their adjacency matrices have the same
eigenvalues. Recently in [http://arxiv.org/pdf/1310.6529v1.pdf] it is proved
that if is any graph cospectral with , then .
In this note, we give a proof of special case of the latter: Any connected
graph cospectral with is isomorphic to . Our proof is independent of
ones given in [http://arxiv.org/pdf/1310.6529v1.pdf] and the proofs are based
on our recent results given in [Trans. Com., 2 no. 4 (2013) 37-52.] Using an
upper bound for the largest eigenvalue of a connected graph given in [J.
Combinatorial Theory, Ser. B, 81 (2001) 177-183.].Comment: 3 pages, 2 figure
On the second largest eigenvalue of the signless Laplacian
Let be a graph of order and let be
the eigenvalues of the -matrix of , also known as the signless Laplacian
of In this paper we give a necessary and sufficient condition for the
equality where In particular, this result solves
an open problem raised by Wang, Belardo, Huang and Borovicanin.
We also show that [ q_{2}(G) \geq\delta(G)] and determine that equality holds
if and only if is one of the following graphs: a star, a complete regular
multipartite graph, the graph or a complete multipartite graph of
the type .Comment: This version fills a gap in one proof, noticed by Rundan Xin
Graphs cospectral with multicone graphs KW 5 L(P)
E. R. van Dam and W. H. Haemers [15] conjectured that almost all graphs are determined by their spectra. Nevertheless, the set of graphs which are known to be determined by their spectra is small. Hence, discovering infinite classes of graphs that are determined by their spectra can be an interesting problem. The aim of this paper is to characterize new classes of multicone graphs that are determined by their spectrum. A multicone graph is defined to be the join of a clique and a regular graph. It is proved that any graph cospectral with multicone graph Kw 5 L(P) is determined by its adjacency spectrum as well as its Laplacian spectrum, where Kw and L(P) denote a complete graph on w vertices and the line graph of the Petersen graph, respectively. Finally, three problems for further researches are proposed.Publisher's Versio