Wide diameter is an important parameter for measuring the reliability and efficiency of interconnection networks. Diameter with width k of a graph G, k-diameter, is defined as the minimum integer d for which there exist at least k internally disjoint paths of length at most d between any two distinct vertices in G. In this paper, we will discuss the wide diameter of two families of interconnection networks and present the bounds of wide diameter of G (G0,G1,...,Gr-1,L,) , where L=∪r-1 i=1 Mi,i+1,Mi,i+1 is an arbitrary perfect matching between V(Gi) and V(Gi+1), and G(G0,G1, F), where {(ui,vi)︱1≤i≤n}U 1{(uivi+1)︱1≤i≤ n} , ui∈V(G0), vi ∈V( G1). And they are used in practical applications, especially in the distributed and parallel computer networks.
The weighted graphs, where the edge weights are positive numbers, are considered. The authors obtain some lower bounds on the spectral radius and the Laplacian spectral radius of weighted graphs, and characterize the graphs for which the bounds are attained. Moreover, some known lower bounds on the spectral radius and the Laplacian spectral radius of unweighted graphs can be deduced from the bounds.
