On distance-balanced graphs (original) (raw)
Related papers
Some Remarks on the 2-Distance-Balanced Graphs
Communications in Mathematics and Applications, 2017
The aim of this paper is to investigate the notion of 2-distance-balanced graphs as a generalized form of distance-balanced graphs. Furthermore, we introduce a subclass of such graphs so-called strongly 2-distance-balanced graphs and present some related results based on Cartesian and lexicographic products of two graphs.
NOTE ON EDGE DISTANCE-BALANCED GRAPHS
2012
Edge distance-balanced graphs are graphs in which for every edge e = uv the number of edges closer to vertex u than to vertex v is equal to the number of edges closer to v than to u. In this paper, we study this property under some graph operations.
On some problems regarding distance-balanced graphs
2022
A graph Gamma\GammaGamma is said to be distance-balanced if for any edge uvuvuv of Gamma\GammaGamma, the number of vertices closer to uuu than to vvv is equal to the number of vertices closer to vvv than to uuu, and it is called nicely distance-balanced if in addition this number is independent of the chosen edge uvuvuv. A graph Gamma\GammaGamma is said to be strongly distance-balanced if for any edge uvuvuv of Gamma\GammaGamma and any integer kkk, the number of vertices at distance kkk from uuu and at distance k+1k+1k+1 from vvv is equal to the number of vertices at distance k+1k+1k+1 from uuu and at distance kkk from vvv. In this paper we answer an open problem posed by Kutnar and Miklavi\v{c} [European J. Combin. 39 (2014), 57-67] by constructing several infinite families of nonbipartite nicely distance-balanced graphs which are not strongly distance-balanced. We disprove a conjecture regarding characterization of strongly distance-balanced graphs posed by Balakrishnan et al. [European J. Combin. 30 (2009), 1048-1053] by pro...
Strongly distance-balanced graphs and graph products
European Journal of Combinatorics, 2009
A graph G is strongly distance-balanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v) − 1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given.
ON SOME PROPERTIES of EDGE QUASI-DISTANCE-BALANCED GRAPHS
Journal of Mathematical Extension, 2021
For an edge e = uv in a graph G, MGu (e) is introduced as the set all edgesof G that are at shorter distance to u than to v. We say that G is an edgequasi-distance-balanced graph whenever for every arbitrary edge e = uv,there exists a constant λ > 1 such that mGu(e) = λ±1mGv(e). We investigatethat edge quasi-distance-balanced garphs are complete bipartite graphsKm,n with m = n. The aim of this paper is to investigate the notion of cyclesin edge quasi-distance-balanced graphs, and expand some techniquesgeneralizing new outcome that every edge quasi-distance-balanced graphis complete bipartite graph. As well as, it is demontrated that connectedquasi-distance-balanced graph admitting a bridge is not edge quasi-distance-balanced graph.
Metric graph theory and geometry: a survey
Surveys on Discrete and Computational Geometry, 2008
The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l 1-graphs. Several kinds of l 1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the graph classes reported in this survey.
Classification of Nicely Edge Distance-Balanced Graphs
2017
A nonempty graph is called nicely edge distance-balanced (as brief we can say NEDB), whenever there exists a positive integer , such that for any edge say we have: . Which denotes the number of edges laying closer to the vertex than vertex and is defined analogously. In this paper, we study on NEDB graph and it’s basic properties and some operations. Also, we try to classify some families of graphs with related .
Studies in graph theory distance related concepts in graphs
2013
By a graph G = (V,E), we mean a finite undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n = |V | and m = |E| respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [7]. In Chapter 1, we collect some basic definitions and theorems on graphs which are needed for the subsequent chapters. The distance d(u, v) between two vertices u and v of a connected graph G is the length of a shortest u-v path in G. There are several distance related concepts and parameters such as eccentricity, radius, diameter, convexity and metric dimension which have been investigated by several authors in terms of theory and applications. An excellent treatment of various distances and distance related parameters are given in Buckley and Harary [6]. Let G = (V,E) be a graph. Let v ∈ V . The open neighborhood N(v) of a vertex v is the set of vertices adjacent to v. Thus N(v) = {w ∈ V : wv ∈ E}. The closed neighborhood of a vertex v, is the set...
On Balancedness of Some Graph Constructions
Let G be a graph with vertex set V(G) and edge set E(G), and let A = {O, I}. A labeling f: V(G) ~ A induces a partial edge labeling f* : E(G) ~ A defined by f*(xy) = f(x), ifand only if f(x) = fey), for each edge xy E E(G). For i E A, let vrt:i) = card{v E V(G): ftv) = i} and ep(i) = card{e E E(G): f*(e) i}. A labeling f ofa graph G is said to be friendly if I vrt:O) - vrt: I) 1 s: I. If 1 ep(O) ef*( 1) 1 s: 1 then G is said to be balanced. Balancedness of the Cartesian product and composition ofgraphs is studied in (19). We provide some new families of balanced graphs using other constructions.
Structural Analysis of Complex Networks, 2010
The distance between two vertices is the basis of the definition of several graph parameters including diameter, radius, average distance and metric dimension. These invariants are examined, especially how they relate to one another and to other graph invariants and their behaviour in certain graph classes. We also discuss characterizations of graph classes described in terms of distance or shortest paths. Finally, generalizations are considered.