ABSTRACT
All the graphs considered in this paper are finite, undirected, and simple. Undefined notations and terminology will conform to those in [1, 2].
Let G be a graph with vertex set V (G) and edge set E(G). Let A(G) be the (0, 1)-adjacency matrix of graph G. The Laplace matrix and the signless Laplace matrix of G is defined to be L(G) = D(G) − A(G) andQ(G) =D(G) + A(G), respectively, where D(G) is the diagonal matrix of vertex degrees of G. We call PG(λ) = |λI − A(G)|, LG(λ) = |λI − L(G)| and QG(λ) = |λI − Q(G)|, respectively, the characteristic polynomial of G with respect to the adjacency matrix, the Laplace matrix and the signless Laplace matrix of graph G, see [2, 3]. Let λ1,λ2 . . . ,λn be n distinct Laplace eigenvalues with the corresponding multiplicities k1, k2 . . . , kn. We denote by
the Laplace spectrum of graph G. GraphG1 ∪ G2 denotes disjoint union ofG1 andG2.
The graphG1 ∨ G2 obtained by connecting each vertex of the graph G1 to each vertex of the graph. G2 · Pn, Cn, and Sn respectively, represent the path, cycle, and star of order n. Call graph P1 ∨ Cn for the umbrella graph with n+ 1 vertices. Let the graph (Pl1 ∪ Pl2 ∪· · · ∪ Pls ) ∨ Cn with be the multiple umbrella graph with n+ s vertices, denoted by Ul1,l2,...,lS ,n.
