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

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.

In this paper, we determine the graph with maximal signless Laplacian spectral radius among all connected graphs with fixed order and given number of cut vertices.