Let F be a finite field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n× n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with pt elements, p = 2, it is shown that mr(F,G) ≤ 2 if and only if the compl...

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s, k)-polar if there exists a partition A, B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i....

Let r(G) be the minimum number of complete bipartite subgraphs needed to partition the edges of G, and let r'G) be the larger of the number of positive and number of negative eigenvalues of G. It is known that T{G) > r(G); graphs with t(G) = r(G) are called eigensharp. Eigensharp graphs include graphs, trees, cycles Cn with n = 4 or n ^ 4k, prisms Cn\2K2 with n ^ 3fc, "twisted prisms" (also cal...

We consider computational problems on covering graphs with bicliques (complete bipartite subgraphs). Given a graph and an integer k, the biclique cover problem asks whether the edge-set of the graph can be covered with at most k bicliques; the biclique partition problem is defined similarly with the additional condition that the bicliques are required to be mutually edge-disjoint. The biclique ...

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose soluti...

The minimum crossing number problem is among the oldest and most fundamental problems arising in the area of automatic graph drawing. In this paper, eight population-based meta-heuristic algorithms are utilized to tackle the minimum crossing number problem for two special types of graphs, namely complete graphs and complete bipartite graphs. A 2-page book drawing representation is employed for ...

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete proble...

‎the concept of the bipartite divisor graph for integer subsets has been considered in [‎m‎. ‎a‎. ‎iranmanesh and c‎. ‎e‎. ‎praeger‎, ‎bipartite divisor graphs for integer subsets‎, ‎{em graphs combin.}‎, ‎{bf 26} (2010) 95--105‎.]‎. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g...

Quasi-random graphs have the property that the densities of almost all pairs of large subsets of vertices are similar, and therefore we cannot expect too large empty or complete bipartite induced subgraphs in these graphs. In this paper we answer the question what is the largest possible size of such subgraphs. As an application, a degree condition that guarantees the connection by short paths ...