The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree ∆ and diameterD. Bipartite graphs of maximum degree ∆, diameterD and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D = 2, 3, 4 and 6, and for D = 3, 4, 6, they have been constructed only for those values of ∆ such that ∆− 1 is ...

It is proved that, in a minimal n-extendable bipartite graph, the subgraph induced by the edges both ends of which have degree at least n + 2 is a forest. As a consequence, every minimal n-extendable bipartite graph has at least 2n + 2 vertices of degree n + 1. This result is sharp. Some other structural results on minimally n-extendable bipartite graphs are also given. @ 1999 Elsevier Science ...

In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipart...

It is known that the Moore bipartite bound provides an upper bound on the order of a connected bipartite graph. In this paper we deal with bipartite graphs of maximum degree ∆ ≥ 2, diameter D ≥ 2 and defect 2 (having 2 vertices less than the Moore bipartite bound). We call such graphs bipartite (∆, D,−2)-graphs. We find that the eigenvalues other than ±∆ of such graphs are the roots of the poly...

A signed bipartite graph G(U, V ) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V ) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V ) is called its signed degree set. In this paper, we prove that every set ...

A graph is called perfect matching compact (briefly, PM -compact), if its perfect matching graph is complete. Matching-covered PM -compact bipartite graphs have been characterized. In this paper, we show that any PM -compact bipartite graph G with δ(G) ≥ 2 has an ear decomposition such that each graph in the decomposition sequence is also PM -compact, which implies that G is matching-covered.

For a graph G of order n and with eigenvalues λ1 > · · · > λn, the HL-index R(G) is defined as R(G) = max { |λb(n+1)/2c|, |λd(n+1)/2e| } . We show that for every connected bipartite graph G with maximum degree ∆ > 3, R(G) 6 √ ∆− 2 unless G is the the incidence graph of a projective plane of order ∆− 1. We also present an approach through graph covering to construct infinite families of bipartit...

We show that the number of independent sets in an N -vertex, d-regular graph is at most (2−1), where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also ...

We shall consider the problem of embedding the complete bipartite graph, Km,n, onto a linear and cyclic chassis in such a way as to minimize the cutwidth. The linear cutwidth of the complete bipartite graph is established and a partial solution to the cyclic cutwidth is presented. It is known that there is a paper in existance [3] that has established the linear cutwidth of the complete biparti...

We generalize the concept of a 2-coloring of a graph to what we call a semibalanced coloring by relaxing a certain discrepancy condition on the shortest-paths hypergraph of the graph. Let G be an undirected, unweighted, connected graph with n vertices and m edges. We prove that the number of different semi-balanced colorings of G is: (1) at most nþ 1 if G is bipartite; (2) at most m if G is non...