Let Wn = K1 ∨ Cn−1 be the wheel graph on n vertices, and let S(n, c, k) be the graph on n vertices obtained by attaching n− 2c− 2k − 1 pendant edges together with k hanging paths of length two at vertex v0, where v0 is the unique common vertex of c triangles. In this paper we show that S(n, c, k) (c > 1, k > 1) and Wn are determined by their signless Laplacian spectra, respectively. Moreover, w...

Let G = (V, E) be a graph without isolated vertices. A set D ⊆ V is a d-distance paired-dominating set of G if D is a d-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a d-distance paired-dominating set for graph G is the d-distance paired-domination number, denoted by γd p(G). In this paper, we study the ddistance paired-domination n...

For a simple graph G, let e(G) denote the number of edges and Sk(G) denote the sum of the k largest eigenvalues of the signless Laplacian matrix of G. We conjecture that for any graph G with n vertices, Sk(G) ≤ e(G) + k+1 2 for k = 1, . . . , n. We prove the conjecture for k = 2 for any graph, and for all k for regular graphs. The conjecture is an analogous to a conjecture by A.E. Brouwer with ...

Let G be a graph of order n with signless Laplacian eigenvalues q1, . . . , qn and Laplacian eigenvalues μ1, . . . , μn. It is proved that for any real number α with 0 < α 6 1 or 2 6 α < 3, the inequality qα 1 + · · · + qα n > μ1 + · · · + μn holds, and for any real number β with 1 < β < 2, the inequality q 1 + · · ·+ q n 6 μβ1 + · · ·+ μ β n holds. In both inequalities, the equality is attaine...

A concept related to the spectrum of a graph is that of energy. The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of G . The Laplacian energy of a graph G is equal to the sum of distances of the Laplacian eigenvalues of G and the average degree d(G) of G. In this paper we introduce the concept of Laplacian energy of fuzzy graphs. ...

Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q1(G) and Laplacian eigenvalues of G by μ1(G) > · · · > μn−1(G) > μn(G) = 0. It is a conjecture on Laplacian spread of graphs that μ1(G)−μn−1(G) 6 n − 1 or equivalently μ1(G) + μ1(G) 6 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, μ1(G)μ1(G) 6 n(n − ...