Let G = ( V , E ) be a simple undirected graph. A labeling f : )→{1, …, k } is local inclusive d -distance vertex irregular of if every adjacent vertices x y ∈ have distinct weights, with the weight w ), sum labels whose distance from at most . The irregularity strength lidis least number for which there exists In this paper, we prove conjecture on 1 tree and generalize result block graph using...

All graphs considered in this paper are finite and undirected, without loops or multiple edges. Let G = (V, E) be a graph. Throughout this paper, let m and n denote the numbers of edges and vertices of graph G, respectively. A connected graph is distance-hereditary if the distance between every two vertices in a connected induced subgraph is the same as that in the original graph. Distance-here...

The distance matrix of a simple graph G is D(G) = (di,j), where di,j is the distance between the ith and jth vertices of G. The greatest eigenvalue λ1 of D(G) is called the distance spectral radius of the graph G and is denoted by λ1(G). A simple connected graph G is called a 2-partite distance regular graph if there exists a partition V1 ∪V2 of the vertex set of G such that for i = 1, 2 and an...

We present a characterization of the linear rank-width of distancehereditary graphs. Using the characterization, we show that the linear rankwidth of every n-vertex distance-hereditary graph can be computed in time O(n · log(n)), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. For our characterization, we combine modifications of canonical spl...

A scan statistic methodology for detecting anomalies has been developed for application to graphs. We equate anomalies with vertices that exhibit high local connectivity properties. In particular we look for cases where all vertices have similar local connectivity, except for one vertex (a champion) that has much higher connectivity at a certain level. For example, a neighborhood champion is a ...

We prove that the out-distance sequence {f(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f(k + 1) ≤ f(k) for k ≥ 1, where f(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertextransitive undirected graph of infinite degree d, we have f(k) = d for all k, 1 ≤ k < diam(G). This answers a quest...

Let G = (V,E) be a graph. The distance between two vertices u and v in a connected graph G is the length of the shortest (u−v) path in G. A set D ⊆ V (G) is a dominating set if every vertex of G is at distance at most 1 from an element of D. The domination number of G is the minimum cardinality of a dominating set of G. A set D ⊆ V (G) is a 2-distance dominating set if every vertex of G is at d...

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in arbitrary fixed dimension, we present an algorithm for this problem that returns in time O(mn log(mn)) the Fréchet distance lower ...

In this note we examine the connection between vertices of high eccentricity and the existence of k-factors in regular graphs. This leads to new results in the case that the radius of the graph is small (≤ 3), namely that a d-regular graph G has all k-factors, for k|V (G)| even and k ≤ d, if it has at most 2d+2 vertices of eccentricity > 3. In particular, each regular graph G of diameter ≤ 3 ha...

A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particula...