in this paper we construct antimagic labelings of the regular complete multipartite graphs and we also extend the construction to some families of regular graphs.

A multigraph G is divisible by t if its edge set can be partitioned into / subsets, such that the subgraphs (called factors) induced by the subsets are all isomorphic. If G has e(G) edges, then it is t-rational if it is divisible by t or if / does not divide e{G). A short proof is given that any graph G is /-rational for all t ^ x'(G) ( t n e chromatic index of G), and thus any r-regular graph ...

An antimagic labeling of a graph G is bijection f:E(G)?{1,…,|E(G)|} such that the weights w(x)=?y?xf(y) distinguish all vertices. A well-known conjecture Hartsfield and Ringel (1990) every connected other than K2 admits an labeling. For set distances D, D-antimagic f:V(G)?{1,…,|V(G)|} weight?(x)=?y?ND(x)f(y) distinct for each vertex x, where ND(x)={y?V(G)|d(x,y)?D} D-neigbourhood x. If ND(x)=r,...

The critical group K(G) of a graph G is a finite abelian group whose order is the number of spanning forests of the graph. Here we investigate the relationship between the critical group of a regular bipartite graph G and its line graph lineG. The relationship between the two is known completely for regular nonbipartite graphs. We compute the critical group of a graph closely related to the com...

The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set ofG can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an open problem in A. Ganesan et al. (2012) [3] about the complexity of determining the regular number of graphs. We show that computation of the regular number for...

let $g$ be a simple graph with an orientation $sigma$‎, ‎which ‎assigns to each edge a direction so that $g^sigma$ becomes a‎ ‎directed graph‎. ‎$g$ is said to be the underlying graph of the‎ ‎directed graph $g^sigma$‎. ‎in this paper‎, ‎we define a weighted skew‎ ‎adjacency matrix with rand'c weight‎, ‎the skew randi'c matrix ${bf‎ ‎r_s}(g^sigma)$‎, ‎of $g^sigma$ as the real skew symmetric mat...

A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs Kn,n, where n = 2. The method involves groups G which factorise as a product XY of two cyclic groups of order ...

We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to measure how much a given graph “resembles” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which ex...

Let us consider a connected graph G with diameter D. For a given integer k between 0 and D, we say that G is k-walk-regular if the number of walks of length between vertices u, v only depends on the distance between u and v, provided that such a distance does not exceed k. Thus, in particular, a 0-walk-regular graph is the same as a walk-regular graph, where the number of cycles of length roote...

Let G be a connected semisimple algebraic group over an algebraically closed field of characteristic zero. A canonical presentation by generators and relations of the algebra of regular functions on G is found. The conjectures of D.Flath and J.Towber, [FT], are proved. (1) Let G be a connected semisimple algebraic group over an algebraically closed field k of characteristic zero. We obtain here...