Using eigenvalue analysis, it was shown by Erdös et al. that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d-1 vertices do not exist for most values of d, when d is even, and we conjecture that they do not exist for any even d greater than 4.

Mixed fault diameter of a graph G, D(a,b)(G), is the maximal diameter of G after deletion of any a vertices and any b edges. Special cases are the (vertex) fault diameter D a = D(a,0) and the edge fault diameter D E a = D(0,a). Let G be a Cartesian graph bundle with fibre F over the base graph B. We show that (1) D a+b+1(G) ≤ D V a (F )+D V b (B) when the graphs F and B are kF -connected and kB...

Mixed fault diameter of a graph G, D(a,b)(G), is the maximal diameter of G after deletion of any a vertices and any b edges. Special cases are the (vertex) fault diameter D a = D(a,0) and the edge fault diameter D E a = D(0,a). Let G be a Cartesian graph bundle with fibre F over the base graph B. We show that (1) D a+b+1(G) ≤ D V a (F )+D V b (B) when the graphs F and B are kF -connected and kB...

We present a study on the relationship between ratio of depth crater to its diameter and for lunar craters both maria highlands. consider younger than 1.1 billion years in age, i.e. Copernican period. The aim this work is improve our understanding such relationships based new estimates craters's diameter. Previous studies considered similar much older (up 3.2 years). calculated depths with diam...

We study the diameter, or the mean distance between sites, in a scale-free network, having N sites and degree distribution p(k) proportional, variant k(-lambda), i.e., the probability of having k links outgoing from a site. In contrast to the diameter of regular random networks or small-world networks, which is known to be d approximately ln(N, we show, using analytical arguments, that scale-fr...

Let denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, . . . , θD is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence θi−1 − βθi + θi+1 = 0 (0 < i < D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have conjectured that q is real if the diameter D is suffi...

An orientation of a digraph D is a spanning subdigraph of D obtained from D by deleting exactly one arc between x and y for every pair x 6= y of vertices such that both xy and yx are in D. In this paper, we consider certain well-known classes of strong digraphs, each member D of which has an orientation with diameter not exceeding the diameter of D by more than a small constant.

Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D 5 21 1, then G has maximum connectivity ( K = 6 ) . and if D 5 21, then it attains maximum edge-connectivity ( A = 6 ) , where I is a parameter which...