275 research outputs found
Proximity and Remoteness in Graphs: a survey
The proximity of a connected graph is the minimum, over
all vertices, of the average distance from a vertex to all others. Similarly,
the maximum is called the remoteness and denoted by . The
concepts of proximity and remoteness, first defined in 2006, attracted the
attention of several researchers in Graph Theory. Their investigation led to a
considerable number of publications. In this paper, we present a survey of the
research work.Comment: arXiv admin note: substantial text overlap with arXiv:1204.1184 by
other author
Wiener index in graphs given girth, minimum, and maximum degrees
Let be a connected graph of order . The Wiener index of is the sum of the distances between all unordered pairs of vertices of . The well-known upper bound on the Wiener index of a graph of order and minimum degree by Kouider and Winkler \cite{Kouider} was improved significantly by Alochukwu and Dankelmann \cite{Alex} for graphs containing a vertex of large degree to . In this paper, we give upper bounds on the Wiener index of in terms of order and girth , where is a function of both the minimum degree and maximum degree . Our result provides a generalisation for these previous bounds for any graph of girth . In addition, we construct graphs to show that, if for given , there exists a Moore graph of minimum degree , maximum degree and girth , then the bounds are asymptotically sharp
Aspects of distance measures in graphs.
Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation
for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack
and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. In Chapter 3, we prove that, if G is a 3-connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4- and 5-connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G
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