For two vertices u, v in a strong digraph D, the strong distance sd(u, v) between u and v is the minimum size of a strong subdigraph of D containing u and v. The upper (lower) orientable strong radius SRAD(G) of a graph G is the maximum (minimum) strong radius over all strong orientations of G. The upper (lower) orientable strong diameter SDIAM(G) of a graph G is the maximum (minimum) strong di...

Lel G be a graph with vertex connectivi2y k(G). An important measure of the fault tolerance of G is its faul&diameler df(G), which is defined as the maximum diameter resulting from the deletion of any set of nodes containing less than k(G) nodes. The robustness of G is often measured by comparing d/(G) with the diameter of Ihe fault-free G, namely d(G). In particular, a family of graphs G, is d...

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D , and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p =...

The purpose of this paper is to prove that, with the exception of C 4 , there are no graphs of diameter 2 and maximum degree d with d 2 vertices . On one hand our paper is an extension of [4] where it was proved that there are at most four Moore graphs of diameter 2 (i .e . graphs of diameter 2, maximum degree d, and d2 + 1 vertices) . We also use the eigenvalue method developed in that paper ....

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d, `) and it is derived from the...

A graph has diameter D if every pair of vertices are connected by a path of at most D edges. The Diameter-D Augmentation problem asks how to add the a number of edges to a graph in order to make the resulting graph have diameter D. It was previously known that this problem is NP-hard [2], even in the D = 2 case. In this note, we give a simpler reduction to arrive at this fact and show that this...

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated lin...

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphsMCK(d, ) and it is derived from the Kautz...

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 prove a number of structural properties of regular graphs of diameter 2, maximum degree d and order d − 1.