The degree of the eigenvalues of generalized Moore geometries

AbstractUsing elementary methods it is proved that the eigenvalues of generalized Moore geometries of type GMm(s, t, c) are of degree at most 3 with respect to the field of rational numbers, if st > 1

Bus interconnection networks

AbstractIn bus interconnection networks every bus provides a communication medium between a set of processors. These networks are modeled by hypergraphs where vertices represent the processors and edges represent the buses. We survey the results obtained on the construction methods that connect a large number of processors in a bus network with given maximum processor degree Δ, maximum bus size r, and network diameter D. (In hypergraph terminology this problem is known as the (Δ,D, r)-hypergraph problem.)The problem for point-to-point networks (the case r = 2) has been extensively studied in the literature. As a result, several families of networks have been proposed. Some of these point-to-point networks can be used in the construction of bus networks. One approach is to consider the dual of the network. We survey some families of bus networks obtained in this manner. Another approach is to view the point-to-point networks as a special case of the bus networks and to generalize the known constructions to bus networks. We provide a summary of the tools developed in the theory of hypergraphs and directed hypergraphs to handle this approach

On strongly closed subgraphs of highly regular graphs

AbstractA geodetically closed induced subgraph Δ of a graph Γ is defined to be strongly closed if Γi(α) ∩ Γ1(β) stays in Δ for every i and α, β ϵ Δ with ∂(α, β) = i. We study the existence conditions of strongly closed subgraphs in highly regular graphs such as distance-regular graphs or distance-biregular graphs

The degree-diameter problem for claw-free graphs and hypergraphs

Please read abstract in the article.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-01182015-08-30hb201

Geodetic Graphs and Convexity.

A graph is geodetic if each two vertices are joined by a unique shortest path. The problem of characterizing such graphs was posed by Ore in 1962; although the geodetic graphs of diameter two have been described and classified by Stemple and Kantor, little is known of the structure of geodetic graphs in general. In this work, geodetic graphs are studied in the context of convexity in graphs: for a suitable family (PI) of paths in a graph G, an induced subgraph H of G is defined to be (PI)-convex if the vertex-set of H includes all vertices of G lying on paths in (PI) joining two vertices of H. Then G is (PI)-geodetic if each (PI)-convex hull of two vertices is a path. For the family (GAMMA) of geodesics (shortest paths) in G, the (GAMMA)-geodetic graphs are exactly the geodetic graphs of the original definition. For various families (PI), the (PI)-geodetic graphs are characterized. The central results concern the family (UPSILON) of chordless paths of length no greater than the diameter; the (UPSILON)-geodetic graphs are called ultrageodetic. For graphs of diameter one or two, the ultrageodetic graphs are exactly the geodetic graphs. A geometry (P,L,F) consists of an arbitrary set P, an arbitrary set L, and a set F (L-HOOK EQ) P x L. The point-flag graph of a geometry is defined here to be the graph with vertex-set P (UNION) F whose edges are the pairs {p,(p,1)} and {(p,1),(q,1)} with p,q (ELEM) P, 1 (ELEM) L, and (p,1),(q,1) (ELEM) F. With the aid of the Feit-Higman theorem on the nonexistence of generalized polygons and the collected results of Fuglister, Damerell-Georgiacodis, and Damerell on the nonexistence of Moore geometries, it is shown that two-connected ultrageodetic graphs of diameter greater than two are precisely the graphs obtained via the subdivision, with a constant number of new vertices, either of all of the edges incident with a single vertex in a complete graph, or of all edges of the form {p,(p,1)} in the point-flag graph of a finite projective plane