Distance Regular Antipodal Covers Of Strongly Regular Graphs
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Antipodal covers
A connected graph $G$ is called distance-regular graph if for any two vertices $u$ and $v$ of $G$ at distance $i$, the number $c_i$ (resp. $B-i$) of neighbours of $v$ at distance $i-1$ (resp. $i+1$) from $u$ depends only on $i$ rather than on individual vertices. A distance regular graph $G$ of diameter $d \in \{2m,2m+1\}$ is antipodal ($r$-cover of its folded graph) if and only if $b_i = c_{d-i}$, for $i = 0,...,d$, $i \ne m$ (and $r = 1+b_m/c_{d-m}$). For example, the dodecahedron is a double-cover of the Petersen graph, the cube is the double-cover of the tetrahedron. Most finite objects of sufficient regularity are closed related to certain distance-regular graphs, in particular, antipodal distance-regular graphs give rise to projective planes, Hadamard matrices and other interesting combinatorial objects. Distance-regular graphs serve as an alternative approach to these objects and allow the use of graph eigenvalues, graph representations, association schemes and the theory of orthogonal polynomials. We start with cyclic covers and spreads of generalized quadrangles and find a switching which uses some known infinite families of antipodal distance-regular graphs of diameter three to produce new ones. Then we examine antipodal distance-families of antipodal distance-regular graphs of diameter three to produce new ones. Then we examine antipodal distance-regular graphs of diameter four and five. Together with J. Koolen we use representations of graphs to extend in the case of diameter four the result P. Terwilliger who has shown, using the theory of Krein modules, that in a $Q$-polynomial antipodal diatance-regular graph the neighbourhood of any vertex is a strongly regular graph. New nonexistence conditions for covers are derived from that. This study relates to the above switching and to extended generalized quadrangles. In an imprimitive association scheme there always exists a merging (i.e., a grouping of the relations) which gives a new nontrivial association scheme. We determine when merging in an antipodal distance-regular graph produces a distance-regular graphs with regular near polygons containing a spread. In case of diameter three we get a Brouwer's characterization of certain distance-regular graphs with generalized quadrangles containing a spread. Finally, antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. In most cases the structure of short cycles provides a tool to determine the existence of an antipodal cover. A relationship between antipodal covers of a graph and its line graph is investigated. Antipodal covers of complete bipartite graphs and their line graphs (lattice graphs) are characterized in terms of weak resolvable transversal designs which are, in the case of maximal covering index, equivalent to affine planes with a parallel class deleted. We conclude by mentioning two results which indicate the importance of antipodal distance-regular graphs. The first one is a joint work with M. Araya and A. Hiraki. Let $G$ be a distance-regular graph of diameter $d$ and valency $k>2$. If $b_t=1$ and $2t \le d$, then $G$ is an antipodal double-cover. Consenquently, if $m>2$ the multiplicity of an eigenvalue of the adjacency matrix of $G$ and if $G$ is not an antipodal double-cover then $d \le 2m-3$. This result is an improvement of Godsil's diameter bound and it is very important for the classification of distance-regular graphs with an eigenvalue of small multiplicity (as opposed to dual classification of distance-regular graphs with small valency). The second result is joint work with C. Godsil. We show that distance-regular graphs that contain maximal independent geodesic paths of short length are antipodal. A new infinite family of feasible parameters of antipodal distance-regular graphs of diameter four is found. As an auxiliary result we use equitable partitions to show that the determinant of a Töplitz matrix can be written as a product of two determinants of approximately half size of the original one.
Strongly Regular Graphs
This monograph on strongly regular graphs is an invaluable reference for anybody working in algebraic combinatorics.
Distance Regular Antipodal Covers of Strongly Regular Graphs
Author: Chris D. Godsil
language: en
Publisher: Faculty of Mathematics, University of Waterloo
Release Date: 1990
We use geometric properties of antipodal distance regular covers to show that infinite families of strongly regular graphs (Steiner graphs and latin square graphs) have no antipodal distance regular covers.