New mixed Moore graphs and directed strongly regular graphs

Jørgensen, Leif K.

doi:10.1016/j.disc.2015.01.013

Cited by 25 publications

(43 citation statements)

References 13 publications

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“…, 6} is a good partition leading to the existence of a family of DSRGs with parameter set (63j + 21, 21j + 6, 7j + 2, 7j + 1, 7j + 2). The corresponding Cayley graphs are DSRGs with parameters (24, 8, 3, 2, 3), (24,9,7,2,4), and (24,10,8,4,4), respectively.…”

Section: Jørgensen's Sporadic Examplesmentioning

confidence: 99%

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Infinite families of directed strongly regular graphs using equitable partitions

Gyürki

2016

Discrete Mathematics

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In this paper we introduce a construction of directed strongly regular graphs from smaller ones using equitable partitions. Each equitable partition of a single DSRG satisfying several conditions leads to an infinite family of directed strongly regular graphs. We construct in this way dozens of infinite families. For order at most 110, we confirm the existence of DSRGs for 30 previously open parameter sets.

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Section: Jørgensen's Sporadic Examplesmentioning

confidence: 99%

“…For DSRG(24,8,3,2,3) the equation (eq1) has only one solution (a, b) = (3, 8), while for (24,9,7,2,4) and (24,10,8,4,4) there are two solutions: (2,12) and (3,8).…”

Section: Jørgensen's Sporadic Examplesmentioning

confidence: 99%

Infinite families of directed strongly regular graphs using equitable partitions

Gyürki

2016

Discrete Mathematics

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“…There is a mixed Moore graph with diameter k = 2 and undirected degree r = 1 for every value of the out-degree z; this graph is obtained by collapsing digons in the Kautz digraphs into edges [21]. Otherwise, just three sporadic mixed Moore graphs have been identified [4,14]. For diameter k = 2, a strong necessary condition on the parameters r and z follows from analysis of the eigenvalues of the graph's adjacency matrix [4].…”

Section: Introductionmentioning

confidence: 99%

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

Tuite

Erskine

2019

Graphs and Combinatorics

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The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree ≤ r and maximum directed out-degree ≤ z. It is also of interest to find smallest possible k-geodetic mixed graphs with minimum undirected degree ≥ r and minimum directed out-degree ≥ z. A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for k = 2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For k = 2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of López and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.

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“…For k = 2, the known examples [3] are a family of Kautz graphs with r = 1, z ≥ 1 and a graph of Bosák with r = 3, z = 1. Recently, Jørgensen [6] has discovered a pair of graphs with r = 3, z = 7.…”

Section: Introductionmentioning

confidence: 99%

A revised Moore bound for mixed graphs

Buset

Amiri

Erskine

et al. 2016

Discrete Mathematics

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The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound ) are extremely rare, but much activity is focussed on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible.There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.

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New mixed Moore graphs and directed strongly regular graphs

Cited by 25 publications

References 13 publications

Infinite families of directed strongly regular graphs using equitable partitions

Infinite families of directed strongly regular graphs using equitable partitions

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

A revised Moore bound for mixed graphs

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