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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Anita Abildgaard Sillasen
Aalborg University
Electrical & Electronics Engineering

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Subdigraphs of almost Moore digraphs induced by fixpoints of an automorphism Anita Abildgaard Sillasen
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 3, No 1 (2015): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2015.3.1.1

Abstract

The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree d and diameter k. An upper bound is given by the Moore bound M(d,k)=1+d+d^2+...+d^k$ and almost Moore digraphs are digraphs with maximum out-degree d, diameter k and order M(d,k)-1. In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism varphi. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregular k-geodetic digraph of degree d'<d-1, order M(d',k)+1 and diameter k+1. As it is known that almost Moore digraphs have an automorphism r, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to r. We determine this for d=4 and d=5, where we prove that except in some special cases, all vertices will have the same order.
On k-geodetic digraphs with excess one Anita Abildgaard Sillasen
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 2, No 2 (2014): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2014.2.2.7

Abstract

A k-geodetic digraph G is a digraph in which, for every pair of vertices u and v (not necessarily distinct), there is at most one walk of length at most k from u to v. If the diameter of G is k, we say that G is strongly geodetic. Let N(d,k) be the smallest possible order for a k-geodetic digraph of minimum out-degree d, then N(d,k) is at most 1+d+d^2+...+d^k=M(d,k), where M(d,k) is the Moore bound obtained if and only if G is strongly geodetic. Thus strongly geodetic digraphs only exist for d=1 or k=1, hence for d,k >1 we wish to determine if N(d,k)=M(d,k)+1 is possible. A k-geodetic digraph with minimum out-degree d and order M(d,k)+1 is denoted as a (d,k,1)-digraph or said to have excess 1.In this paper we will prove that a (d,k,1)-digraph is always out-regular and that if it is not in-regular, then it must have 2 vertices of in-degree less than d, d vertices of in-degree d+1 and the remaining vertices will have in-degree d.Furthermore we will prove there exist no (2,2,1)-digraphs and no diregular (2,k,1)-digraphs for k> 2.
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