Nonexistence of certain cubic graphs with small diameters
Discrete Mathematics - Special issue on combinatorics and algorithms
The Mathematica book (4th edition)
The Mathematica book (4th edition)
Interconnection Networks: An Engineering Approach
Interconnection Networks: An Engineering Approach
Principles and Practices of Interconnection Networks
Principles and Practices of Interconnection Networks
On bipartite graphs of defect 2
European Journal of Combinatorics
On bipartite graphs of diameter 3 and defect 2
Journal of Graph Theory
Topological Structure and Analysis of Interconnection Networks
Topological Structure and Analysis of Interconnection Networks
On bipartite graphs of defect at most 4
Discrete Applied Mathematics
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The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Δ and diameter D. Bipartite graphs of maximum degree Δ, diameter D 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 Δ such that Δ驴1 is a prime power.The scarcity of Moore bipartite graphs, together with the applications of such large topologies in the design of interconnection networks, prompted us to investigate what happens when the order of bipartite graphs misses the Moore bipartite bound by a small number of vertices. In this direction the first class of graphs to be studied is naturally the class of bipartite graphs of maximum degree Δ, diameter D, and two vertices less than the Moore bipartite bound (defect 2), that is, bipartite (Δ,D,驴2)-graphs.For Δ驴3 bipartite (Δ,2,驴2)-graphs are the complete bipartite graphs with partite sets of orders Δ and Δ驴2. In this paper we consider bipartite (Δ,D,驴2)-graphs for Δ驴3 and D驴3. Some necessary conditions for the existence of bipartite (Δ,3,驴2)-graphs for Δ驴3 are already known, as well as the non-existence of bipartite (Δ,D,驴2)-graphs with Δ驴3 and D=4,5,6,8. Furthermore, it had been conjectured that bipartite (Δ,D,驴2)-graphs for Δ驴3 and D驴4 do not exist. Here, using graph spectra techniques, we completely settle this conjecture by proving the non-existence of bipartite (Δ,D,驴2)-graphs for all Δ驴3 and all D驴6.