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An almost Moore ( d, k)-digraph is a regular digraph of degree $${d > 1}$$ , diameter $${k > 1}$$ and order $${N(d, k) = d + d^{2} +\cdots + d^{k}}$$ . So far, their existence has only been showed for k = 2. Their nonexistence has been proved for k = 3, 4 and for d = 2, 3 when $${k \geq 3}$$ . In this paper, we prove that (4, k) and (5, k)-digraphs with self-repeats do not exist for infinitely many primes k. [ABSTRACT FROM AUTHOR]

This paper provides constructions of antimagic labelings of two families of graphs, namely, sequential generalized corona graphs and generalized snowflake graphs. The labelings can also be used for some families of antimagic trees which are special cases of these families. Future lines of research for antimagic graph labelings are proposed. [ABSTRACT FROM AUTHOR]

An antimagic labeling of a graph with q edges is a bijection from the set of edges to the set of positive integers {1, 2, ..., q} such that all vertex weights are pairwise distinct, where the vertex weight of a vertex is the sum of the labels of all edges incident with that vertex. A graph is antimagic if it has an antimagic labeling. In this paper, we provide antimagic labelings for a family of generalized pyramid graphs. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we study magic labeling of type (1, 1, 1) for wheels and subdivided wheels. Secondly, we prove that a wheel admits a magic labeling of type (0, 1, 1). We also prove that any odd wheel admits a magic labeling of type (0, 1, 0). At the last, we find the magic labeling of type (1, 1, 0) for subdivided wheels. [ABSTRACT FROM AUTHOR]

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each $${k\geq 2,\,q\geq\binom{k+1}{2}}$$ with k|2 q, there exists an antimagic k-regular graph with q edges and p = 2 q/ k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examines the existence of such labelings for disjoint union of plane graphs. [ABSTRACT FROM AUTHOR]

By the extremal number ex( n; t) = ex( n; { C3, C4, . . . , C t}) we denote the maximum size (that is, number of edges) in a graph of order n > t and girth at least g ≥ t + 1. The set of all the graphs of order n, containing no cycles of length ≥ t, and of size ex( n; t), is denoted by EX( n; t) = EX( n; { C3, C4, . . . , C t}), these graphs are called EX graphs. In 1975, Erdős proposed the problem of determining the extremal numbers ex( n; 4) of a graph of order n and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5. In particular, we prove that ex(29; 6) = 45, also we improve some lower bounds and upper bounds of ex u( n; t), for some particular values of n and t. [ABSTRACT FROM AUTHOR]

The Moore bound for a diregular digraph of degree E5>, k and diameter k is . It is known that digraphs of order do not exist for d>1 and k>1 ([24] or [6]). In this paper we study digraphs of degree E5>, k, diameter k and order , denoted by ( d, k)- digraphs. Miller and Fris showed that (2, k)-digraphs do not exist for k≥3 [22]. Subsequently, we gave a necessary condition of the existence of (3, k)-digraphs, namely, (3, k)-digraphs do not exist if k is odd or if k+1 does not divide [3]. The (E5>, k, 2)-digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of ( d, k)-digraphs. In particular, for , we show that a ( d, k)-digraph contains either no cycle of length k or exactly one cycle of length k. [ABSTRACT FROM AUTHOR]