Let G be a nontrivial connected graph. The distance between two vertices u and v of G is the length of a shortest u-v path in G. Let u be a vertex in G. A vertex v is an eccentric vertex of u if d(u, v) = e(u), that is every vertex at greatest distance from u is an eccentric vertex of u. A vertex v is an eccentric vertex of G if v is an eccentric vertex of some vertex of G. Consequently, if v i...

‎The textit{metric dimension} of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$‎. ‎In this case‎, ‎$B$ is called a textit{metric basis} for $G$‎. ‎The textit{basic distance} of a metric two dimensional graph $G$ is the distance between the elements of $B$‎. ‎Givi...

In this paper, we give some sufficient conditions for distance local connectivity of a graph, and a degree condition for local connectivity of a k-connected graph with large diameter. We study some relationships between t-distance chromatic number and distance local connectivity of a graph and give an upper bound on the t-distance chromatic number of a k-connected graph with diameter d.

Let G be a connected graph and S a nonempty set of vertices of G. Then the Steiner distance d,(S) of S is the smallest number of edges in a connected subgraph of G that contains S. Let k, I, s and m be nonnegative integers with m > s > 2 and k and I not both 0. Then a connected graph G is said to be k-vertex I-edge (s,m)-Steiner distance stable, if for every set S of s vertices of G with d,(S) ...

‎the gutman index and degree distance of a connected graph $g$ are defined as‎ ‎begin{eqnarray*}‎ ‎textrm{gut}(g)=sum_{{u,v}subseteq v(g)}d(u)d(v)d_g(u,v)‎, ‎end{eqnarray*}‎ ‎and‎ ‎begin{eqnarray*}‎ ‎dd(g)=sum_{{u,v}subseteq v(g)}(d(u)+d(v))d_g(u,v)‎, ‎end{eqnarray*}‎ ‎respectively‎, ‎where‎ ‎$d(u)$ is the degree of vertex $u$ and $d_g(u,v)$ is the distance between vertices $u$ and $v$‎. ‎in th...

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomialtime algorithm, Algorithmica 78(1):342–377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance-hereditary graphs based on this charac...

Many applications such as information retrieval and classification, involve measuring graph distance or similarity, i.e., matching graphs to identify and quantify their common features. Different kinds of graph matchings have been proposed, giving rise to different graph similarity or distance measures. Graph matchings may be univalent – when each vertex is associated with at most one vertex of...

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 feedba...

A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well a...

Let G be a connected graph with vertex set V (G). The degree resistance distance of G is defined as DR(G) = ∑ fu,vg V (G)[d(u) +d(v)]R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between u and v. In this paper, we characterize n-vertex unicyclic graphs having minimum and second minimum degree resistance distance.