The Unique Mixed Almost Moore Graph with Parameters k = 2, r = 2 and z = 1

Abstract

A natural upper bound for the maximum number of vertices in a mixed graph with maximum undirected degree r, maximum directed out-degree z and diameter k is given by the mixed Moore bound. Graphs with order attaining the Moore bound are known as Moore graphs, and they are very rare. Besides, graphs with prescribed parameters and order one less than the corresponding Moore bound are known as almost Moore graphs. In this paper we prove that there is a unique mixed almost Moore graph of diameter k = 2 and parameters r = 2 and z = 1.

Almost Moore and the largest mixed graphs of diameters two and three

“…Let P (i) , R (i) , and Z (i) the matrices corresponding to the above permutations σ (i) , ρ (i) , and ω (i) , respectively. Let A (4) = R (1) + Z (2) , A (5) = R (2) + Z (3) , A (6) = P (1) + Z (2) , and A (7) = P (2) + Z (3) , which correspond to adjacency matrices of the mixed graphs with parallel arcs H (4) , H (5) , H (6) , H (7) , respectively (see Figure 3). Then, (a) (R (i) ) 2 = I, and A (i) A (j) = A (j) A (i) for all i, j ∈ {1, 2, 3}.…”

“…Although some such Moore mixed graphs of diameter two are known to exist, and they are unique (see Nguyen,Miller,and Gimbert [18]), the general problem remains unsettled. Again for diameter 2, Bosák [2] gave a necessary condition for the existence of a Moore mixed graph, but recently it was proved that there is no Moore mixed graph for the (r, z) pairs (3, 3), (3,4), and (7, 2) satisfying such necessary condition (see López,Miret,and Fernández [17]). In general, there are infinitely many pairs (r, z) satisfying Bosák necessary condition for which the existence of a Moore mixed graph is not known yet.…”

“…, m 10 ) of this graph is m 5 = 2 and m i = 0, for all i = 5. For more information on its existence, see López and Miret [16]; and on its unicity, see Buset, López, and Miret [4]). Moreover, answering an open question of the two last authors [16], Tuite and Erskine [19] proved that (r, z, 2)-and (1, 1, k)-almost Moore mixed graphs are totally regular (that is, all vertices have undirected degree r and out-and in-degree z).…”

On Total Regularity of Mixed Graphs with Order Close to the Moore Bound

“…In [7] it is proven that there is a unique almost mixed Moore graph with diameter k = 2 and degree parameters r = 2, z = 1. At the time it was unknown whether such a graph must be totally regular.…”

“…its undirected subgraph is regular and its directed subgraph is diregular. In [7] it is shown that this is the unique almost mixed Moore graph with these parameters; the proof is complicated by the fact that any such graph could not be assumed to be totally regular. These considerations prompted López and Miret to ask whether almost mixed Moore graphs with diameter two must be totally regular [16].…”