In this paper we compute the dual code of the evaluation codes arising from the incidence matrix of a complete bipartite graph. Mathematics Subject Classification: 94B27, 94B60

a finite group $g$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. taeri conjectured that an insoluble group satisfying this condition is isomorphic to $s times a$ where $a$ is abelian and $s cong psl_2(q)$ for $q in {4,8}$. we confirm this conjecture.

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by lak(G), is the least number of linear k-forests needed to decompose G. In this paper, we completely determine lak(G) when G is a balanced complete bipartite graph Kn,n or a complete graph Kn, and k = 3. © 2007 Elsevier B.V. All rights reserved.

In this paper, a method for obtaining large diameter 6 graphs by replacing some vertices of a Moore bipartite diameter 6 graph with complete Kh graphs is proposed. These complete graphs are joined to each other and to the remaining nonmodified graphs by means of new edges and by using a special diameter 2 graph. The degree of the graph so constructed coincides with the original one. © 1999 John...

We deal with the graph operator Pow2 defined to be the complement of the square of a graph: Pow2(G) = Pow2(G). Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class G of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph Km,n can be deco...

A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for each two colors i and j with i < j, every vertex of G colored by j has a neighbor with color i. The Grundy chromatic number Γ(G), is the largest integer k for which there exists a Grundy k-coloring for G. In this note we first give an interpretation of Γ(G) in terms of the total graph of G, when G is the complement of a...

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...

The Feedback Arc Set problem asks whether it is possible to delete at most k arcs to make a directed graph acyclic. We show that Feedback Arc Set is NPcomplete for bipartite tournaments, that is, directed graphs that are orientations of complete bipartite graphs.