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### Enumerations of vertex orders of almost Moore digraphs with selfrepeats

E.T. Baskoro, Y.M. Cholily, M. Miller
2008 Discrete Mathematics
In this paper, we propose an exact formula for the number of all vertex orders in an almost Moore digraph G containing selfrepeats, based on the vertex orders of the out-neighbours of any selfrepeat vertex  ...  An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices one less than the Moore bound.  ...  However, since the order of a (d, k)-digraph G is one less than the Moore bound then for every vertex u ∈ V (G) there exists exactly one vertex v ∈ V (G) such that there are two walks of length k from  ...

### Enumeration of almost Moore digraphs of diameter two

J. Gimbert
2001 Discrete Mathematics
An almost Moore (d; 2)-digraph is a regular directed graph of degree d ¿ 1, diameter k = 2 and order n one less than the (unattainable) Moore bound.  ...  This allows us to complete the classiÿcation of the almost Moore (d; 2)-digraphs up to isomorphisms.  ...  Poonen for sending me their proof of Theorem 1. Thanks are also due to A. Rio for her useful information. Likewise, I am very grateful to M.A.  ...

### Nonexistence of Almost Moore Digraphs of Diameter Three

J. Conde, J. Gimbert, J. Gonzàlez, J. M. Miret, R. Moreno
2008 Electronic Journal of Combinatorics
Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound \$M(d,k)=1+  ...  The enumeration of almost Moore digraphs of degree \$d\$ and diameter \$k=3\$ turns out to be equivalent to the search of binary matrices \$A\$ fulfilling that \$AJ=dJ\$ and \$I+A+A^2+A^3=J+P\$, where \$J\$ denotes  ...  From now on, regular digraphs of degree d > 1, diameter k > 1 and order n = d + · · · + d k will be called almost Moore (d, k)-digraphs (or (d, k)-digraphs for short).  ...

### On mixed almost Moore graphs of diameter two

Nacho López, Josep Miret
unpublished
Mixed almost Moore graphs appear in the context of the Degree/Diameter problem as a class of extremal mixed graphs, in the sense that their order is one less than the Moore bound for mixed graphs.  ...  In this paper we give some necessary conditions for the existence of mixed almost Moore graphs of diameter two derived from the factorization in Q[x] of their characteristic polynomial.  ...  In the other way around, for the case of directed graphs, Gimbert [8] enumerated all almost Moore digraphs of diameter two.  ...