ON THE EIGENVALUES OF N-CAYLEY GRAPHS: A SURVEY

A graph Γ is called an n-Cayley graph over a group G if Aut(Γ) contains a semi-regular subgroup isomorphic to G with n orbits. In this paper, we review some recent results and future directions around the problem of computing the eigenvalues on n-Cayley graphs

Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge- transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers

Tight factorizations of girth-gg-regular graphs

The determination of 1-factorizations $F$ of girth-regular graphs of girth, regular degree and chromatic index $g$ is proposed for the cases in which each girth cycle intersects every 1-factor of $F$. This endeavor may apply to priority assignment problems, managerial situations in optimization and decision making. Applications to hamiltonian decomposability (via union of pairs of 1-factors) and to 3-dimensional geometry (M\"obius-strip compounds and hollow-triangle polylinks) are given.Comment: 37 pages, 19 figures, 10 tables05C