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Abstract: For a given graph , a non-increasing sequence of nonnegative integers is said to be potentially -graphic if there exists a realization of containing as a subgraph. The split graph on vertices is denoted by . In this paper, we give a Rao-type characterization for to be potentially -graphic. A simplification of this characterization is also presented. [Copyright &y& Elsevier]

Abstract: In this paper, we provide an upper bound for the k-tuple domination number that generalises known upper bounds for the double and triple domination numbers. We prove that for any graph G,where is the k-tuple domination number; is the minimal degree; is the m-degree of G; if or 2 and if ; d is the average degree. [Copyright &y& Elsevier]

Abstract: We find an upper bound on the algebraic connectivity of graphs of various genus. We begin by showing that for fixed k ⩾1, the graph of genus k of largest algebraic connectivity is a complete graph. We then find an upper bound for noncomplete graphs of a fixed genus k ⩾1 and we determine the values of k for which the upper bound can be attained. Finally, we find the upper bound of the algebraic connectivity of planar graphs (graphs of genus zero) and determine precisely which graphs attain this upper bound. [Copyright &y& Elsevier]

Abstract: A graph X is said to be distance-balanced if for any edge of X, the number of vertices closer to u than to is equal to the number of vertices closer to than to u. A graph X is said to be strongly distance-balanced if for any edge of X and any integer k, the number of vertices at distance k from u and at distance from is equal to the number of vertices at distance from u and at distance k from . Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: , , , and . [Copyright &y& Elsevier]

Abstract: The irregularity strength of a simple graph is the smallest integer k for which there exists a weighting of the edges with positive integers at most k such that all the weighted degrees of the vertices are distinct. In this paper we study the irregularity strength of circulant graphs of degree 4. We find the exact value of the strength for a large family of circulant graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we give out a counterexample of the Graffiti''s conjecture (583), and get the best bounds of , where T denotes a tree, denotes the inverse degree of T and matching of T. [Copyright &y& Elsevier]

Abstract: In this note, we characterize the minimally circular-imperfect graphs that have a major vertex. [Copyright &y& Elsevier]

Abstract: Given a connected weighted graph , we consider a hypergraph corresponding to the set of all shortest paths in G. For a given real assignment on V satisfying , a global rounding with respect to is a binary assignment satisfying that for every . We conjecture that there are at most global roundings for , and also the set of global roundings is an affine independent set. We give several positive evidences for the conjecture. [Copyright &y& Elsevier]

Let α denote the average degree, and δ denote the minimum degree of a graph. An edge is light if both its endpoints have degree bounded by a constant depending only on α and δ. A graph is degree-constrained if α<2δ. The primary result of this paper is that every degree-constrained graph has a light edge. Most previous results in this direction have been for embedded graphs. This result is extended in a variety of ways. First it is proved that there exists a constant c(α,δ) such that for every 0&les;&epsiv;, every degree-constrained graph with n vertices has at least &epsiv;·n light edges. An analogous result is proved guaranteeing a matching of light edges. The method is refined in the case of planar graphs to obtain improved degree bounds. [Copyright &y& Elsevier]

It is known that the independence number of a connected claw-free graph G of order n is at most (n+1)/2. We improve this result by showing that this bound still holds for the upper irredundance number IR(G). We characterize the connected claw-free graphs for which IR(G)=(n+1)/2 and give some properties of those graphs for which IR(G)=n/2 if n is even or IR(G)=(n−1)/2 if n is odd. [Copyright &y& Elsevier]

For terminology and notation in graph theory, we refer the reader to B. BOLLOBAS [12]; others will be specified in the text as and when found necessary. The graphs considered in this paper, unless mentioned otherwise, are possibly infinite, simple and without loops. [Copyright &y& Elsevier]