The biclique cover number (resp. biclique partition number) of a graph G, bc(G) (resp. bp(G)), is the least number of bicliques—complete bipartite subgraphs—that are needed to cover (resp. partition) the edges of G. The local biclique cover number (resp. local biclique partition number) of a graph G, lbc(G) (resp. lbp(G)), is the least r such that there is a cover (resp. partition) of the edges...

It is known that determination problem of the bandwidth for an arbitrary graph is NP-complete. This paper establishes the bandwidth for the composition of a complete bipartite graph with other graphs; the strong product of a complete bipartite graph with a complete graph, a path, a cycle; and the tensor product of a complete graph with a complete graph.

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...

Let L be a lattice with the least element 0. An element x ∈ L is a zero divisor if x∧ y = 0 for some y ∈ L∗ = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph Γ(L) to L with vertex set Z(L)∗ = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y = 0. In this paper, we obtain certain properties and dia...

Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove...