For a vertex v of a graph G, a spanning tree T of G is distancepreserving from v if, for any vertex w, the distance from v to w on T is the same as the distance from v to w on G. If two vertices u and v are distinct, then two distance-preserving spanning trees Tu from u and Tv from v are distinct in general. A purpose of this paper is to give a characterization for a given weighted graph G to h...

Vertex k-center problem introduces the notion to recognize k locations as centers in a given network of n connected nodes holding the condition of triangle inequality. This paper presents an efficient algorithm that provides a better solution for vertex k-center problem. Anticipatory Bound Selection Procedure (ABSP) is deployed to find the initial threshold distance (or radius) and eradicating ...

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customar...

The broadcast domination problem is a variant of the classical minimum dominating set problem in which a transmitter of power p at vertex v is capable of dominating (broadcasting to) all vertices within distance p from v. Our goal is to assign a broadcast power f(v) to every vertex v in a graph such that ∑ v∈V f(v) is minimized, and such that every vertex u with f(u) = 0 is within distance f(v)...

A distance-k dominating set S of a directed graph D is a set of vertices such that for every vertex v of D, there is a vertex u ∈ S at distance at most k from it. Minimum distance-k dominating set is especially important in communication networks for distributed data structure and for server placement. In this paper, we shall present simple distributed algorithms for finding the unique minimum ...

For a vertex v and a (k−1)-element subset P of vertices of a graph, one can define the distance from v to P in various ways, including the minimum, average, and maximum distance from v to P . Associated with each of these distances, one can define the keccentricity of the vertex v as the maximum distance over all P , and the k-eccentricity of the set P as the maximum distance over all v. If k =...

‎the second multiplicative zagreb coindex of a simple graph $g$ is‎ ‎defined as‎: ‎$${overline{pi{}}}_2left(gright)=prod_{uvnotin{}e(g)}d_gleft(uright)d_gleft(vright),$$‎ ‎where $d_gleft(uright)$ denotes the degree of the vertex $u$ of‎ ‎$g$‎. ‎in this paper‎, ‎we compare $overline{{pi}}_2$-index with‎ ‎some well-known graph invariants such as the wiener index‎, ‎schultz‎ ‎index‎, ‎eccentric co...

Abstract The dual-cube, derivable from the hypercube, admits a number of good properties that render it as network topology, especially when node size exceeds several million. This paper presents other welcome characteristics graph. Prominent among them are: (1) vertex transitivity facilitates working an algorithm meant for “local” context in global well, (2) exact formula distance between two ...