Let G=(V,A) be a digraph. The eccentricity e(u) of a vertex u is the maximum distance from u to any other vertex in G. A vertex v in G is an eccentric vertex of u if the distance from u to v equals e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G and has arcs from a vertex v to its eccentric vertices. In this paper we present several results on the eccentric digraph...

An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u− v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in t...

Let [Formula: see text] be a graph. A set [Formula: see text] is a distance k-dominating set of G if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text], where k is a positive integer. The distance k-domination number [Formula: see text] of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as ...

Abstract The quad-cube is a special case of the metacube that itself derivable from hypercube. It amenable to an application as network topology, especially when node size exceeds several million. This paper presents following welcome properties graph, relating its structure: (1) vertex transitivity facilitates working algorithm meant for “local” context in global well, and (2) exact formula di...

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library Surmnary. The (direc...

In this paper we show that the θ-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last θ-graph for which it was not known whether its stretch factor was bounded.

A set of cycles C1, . . . , Ck in a graph G is said to identify the vertices v if the sets {j : v ∈ Cj} are all nonempty and di erent. In this paper, bounds for the minimum possible k are given when G is the graph Zp endowed with the Lee or Hamming metric or G is a complete bipartite graph.

Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. The degree resistance distance of a graph G is defined as D R (G) =  {u,v}⊆V (G) [d(u) + d(v)]R(u, v), where d(u) is the degree of the vertex u, and R(u, v) the resistance distance between the vertices u and v. Let Cact(n; t) be the set of all cacti possessing n vertices and t cy...