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Based on a separator theorem for general surfaces we prove a Moore bound for graphs of given degree and diameter, embedded in a fixed surface. The problem of determining the largest order (i.e., number of vertices) n(d, k) of a graph of maximum degree at most d and diameter at most k is well known as the degree-diameter problem. A spanning tree argument… (More)
For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3)k. By comparison with other available results in this area we show that, for all sufficiently large d and k, our family gives the current largest known Cayley graphs of degree d and diameter k.
Let ) 2 , (d C , ) 2 , (d AC , and ) 2 , (d CC be the largest order of a Cayley graph of a group, an Abelian group, and a cyclic group, respectively, of diameter 2 and degree d. The currently known best lower bounds on these parameters are 2 / ) 1 ( ) 2 , ( 2 d d C for degrees 1 2 q d where q is an odd prime power, ) 4 )( 8 / 3 ( ) 2 , ( 2 d d… (More)
In this note we prove a Moore-like bound for graphs of diameter two and given degree which arise as lifts of dipoles with loops and multiple edges, with voltage assignments in Abelian groups.
Consider a composition of two regular coverings π1 : Γ0 → Γ1 and π2 : Γ1 → Γ2 of graphs, given by voltage assignments α1, α2 on Γ1, Γ2 in groups G1 and G2, respectively. In the case when π2 ◦ π1 is regular we present an explicit voltage assignment description of the composition in terms of G1, G2, α1, α2, and walks in Γ1.
In a mixed (Δ, d)-regular graph, every vertex is incident with Δ ≥ 1 undirected edges and there are d ≥ 1 directed edges entering and leaving each vertex. If such a mixed graph has diameter 2, then its order cannot exceed (Δ+ d) + d+1. This quantity generalizes the Moore bounds for diameter 2 in the case of undirected graphs (when d = 0) and digraphs (when… (More)