We determine the graph with the largest signless Laplacian spectral radius among all unicyclic graphs with fixed matching number.

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number ω, then the largest eigenvalue q (G) of the matrix Q = A+D satisfies q (G) ≤ 2 (1− 1/ω)n. If G is a complete regu...

The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.

Detecting protein complexes is an important way to discover the relationship between network topological structure and its functional features in protein-protein interaction (PPI) network. The spectral clustering method is a popular approach. However, how to select its optimal Laplacian matrix is still an open problem. Here, we analyzed the performances of three graph Laplacian matrices (unnorm...

Let U(n, g) and B(n, g) be the set of unicyclic graphs and bicyclic graphs on n vertices with girth g, respectively. Let B1(n, g) be the subclass of B(n, g) consisting of all bicyclic graphs with two edge-disjoint cycles and B2(n, g) = B(n, g)\B1(n, g). This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in U(n, g) and B(n, g), respectivel...

A connected graph is called a c-cyclic graph if it contains n vertices and n + c − 1 edges. Let C(n, k, c) denote the class of connected c-cyclic graphs with n vertices and k pendant vertices. Recently, the unique extremal graph, which has greatest (respectively, signless) Laplacian spectral radius, in C(n, k, c) has been determined for 0 ≤ c ≤ 3, k ≥ 1 and n ≥ 2c + k + 1. In this paper, the un...

In this paper we define extended corona and extended neighborhood corona of two graphs G1 and G2, which are denoted by G1 • G2 and G1 ∗ G2 respectively. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As applications, we give methods to construct infinite families of integral graphs, Laplacian integral graphs and expander graphs from known ones.

A connected graph is called a c-cyclic graph if it contains n vertices and n + c − 1 edges. Let C(n, k, c) denote the class of connected c-cyclic graphs with n vertices and k pendant vertices. Recently, the unique extremal graph, which has greatest (respectively, signless) Laplacian spectral radius, in C(n, k, c) has been determined for 0 ≤ c ≤ 3, k ≥ 1 and n ≥ 2c + k + 1. In this paper, the un...

The signless Laplacian spectral radius of a graph G, denoted by q(G), is the largest eigenvalue its matrix. In this paper, we investigate extremal for graphs without short cycles or long cycles. Let G(m,g) be family on m edges with girth g and H(m,c) circumference c. More precisely, obtain unique maximal q(G) in H(m,c), respectively.