In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determi...

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 ...

This paper presents a novel technique for simplifying a triangulated surface at different levels of resolution. While most existing algorithms, based on iterative vertex decimation, employ the distance for error metric, the proposed algorithm utilizes an edge criterion for removing a vertex. An interior angle of a vertex is defined as the maximum value of all possible angles formed by combinati...

The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v, and u is an eccentric vertex for v if its distance from v is d(u, v) = e(v). A vertex of maximum eccentricity in a graph G is called peripheral, and the set of all such vertices is the peripherian, denoted PeriG). We use Cep(G) to denote the set of eccentric vertices of vertices in C(G). A graph G is called ...

For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedb...

For a given graph G, ε(v) and deg(v) denote the eccentricity and the degree of the vertex v in G, respectively. The adjacent eccentric distance sum index of a graph G is defined as [Formula in text], where [Formula in text] is the sum of all distances from the vertex v. In this paper we derive some bounds for the adjacent eccentric distance sum index in terms of some graph parameters, such as i...

Abstract We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine parameterizations this admits polynomial kernel. While characterization is known related Cover based on recently introduced notion of bridge-depth, it remaine...

In the present paper, we classify abelian antipodal distance-regular graphs \(\Gamma\) of diameter 3 with following property: \((*)\) has a transitive group automorphisms \(\widetilde{G}\) that induces primitive almost simple permutation \(\widetilde{G}^{\Sigma}\) on set \({\Sigma}\) its classes. There are several infinite families (arc-transitive) examples in case when rank \({\rm rk}(\widetil...