We first establish the relationship between the largest eigenvalue of the Laplacian matrix of a graph and its bipartite density. Then we present lower and upper bounds for the largest Laplacian eigenvalue of a graph in terms of its largest degree and diameter.

The largest eigenvalue of a network’s adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of any network to the largest eigenvalue of two network subgraphs, considered as isolated: The ...

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) the Harary (also called matrix) while diag(RH(G)) represents diagonal of total vertices. In present work, some upper and lower bounds second-largest eigenvalue graphs in terms various parameters are investigated. Besides, all attaining these new characterized. Addi...