The smallest graph of girth 6 and valency 7

O'keefe, M.; Wong, Pak-Ken

doi:10.1002/jgt.3190050105

Cited by 31 publications

(22 citation statements)

References 6 publications

(3 reference statements)

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“…§1. It was rediscovered by O'KEEFE & WONG [578] and ITO [427] found that Autf -3·Alt (7). It was rediscovered by O'KEEFE & WONG [578] and ITO [427] found that Autf -3·Alt (7).…”

Section: Cagesmentioning

confidence: 99%

Distance-Regular Graphs

1989

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“…§1. It was rediscovered by O'KEEFE & WONG [578] and ITO [427] found that Autf -3·Alt (7). It was rediscovered by O'KEEFE & WONG [578] and ITO [427] found that Autf -3·Alt (7).…”

Section: Cagesmentioning

confidence: 99%

Distance-Regular Graphs

1989

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“…The question of the construction of graphs with small excess is a difficult one, in the papers [1,[3][4][5]7,10,8,11,13,15,[18][19][20][21][22]24,27,26] the authors focused on constructing the smallest one. More details about constructions on cages can be found in the surveys by Wong [28], by Holton and Sheehan [17] or in the recent survey by Exoo and Jajcay [14].…”

Section: Introductionmentioning

confidence: 99%

On upper bounds of odd girth cages

Araujo‐Pardo

2010

Discrete Mathematics

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a b s t r a c tWe give a construction of k-regular graphs of girth g using only geometrical and combinatorial properties that appear in any (k; g + 1)-cage, a minimal k-regular graph of girth g + 1. In this construction, g ≥ 5 and k ≥ 3 are odd integers, in particular when k − 1 is a power of 2 and (g + 1) ∈ {6, 8, 12} we use the structure of generalized polygons. With this construction we obtain upper bounds for the (k; g)-cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 5, 7, 11.

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“…Cages have been studied intensely since they were introduced by Tutte [21] in 1947. Erdős and Sachs [10] proved the existence of a graph for any value of the regularity k and the girth g, thus most of the work carried out has been focused on constructing a smallest one [1,3,5,8,11,12,18,16,17,23].…”

Section: Introductionmentioning

confidence: 99%

Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices

Balbuena¹

2008

SIAM J. Discrete Math.

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Let q be a prime power and r = 0, 1 . . . , q − 3. Using the Latin squares obtained by multiplying each entry of the addition table of the Galois field of order q by an element distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q − r)-regular bipartite graphs of girth 6 and q 2 − rq − 1 vertices in each partite set. Moreover, in this work two Latin squares of order q − 1 with entries belonging to {0, 1, . . . , q}, not necessarily the same, are defined to be quasi row-disjoint if and only if the cartesian product of any two rows contains at most one pair (x, x) with x = 0. Using these quasi row-disjoint Latin squares we find (q − 1)-regular bipartite graphs of girth 6 with q 2 − q − 2 vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.

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The smallest graph of girth 6 and valency 7

Abstract: With the aid of a computer. we give a regular graph of girth 6 and valency 7, which has 90 vertices and show that this is the unique smallest graph with these properties.

Cited by 31 publications

References 6 publications

Distance-Regular Graphs

Distance-Regular Graphs

On upper bounds of odd girth cages

Incidence Matrices of Projective Planes and of Some Regular Bipartite Graphs of Girth 6 with Few Vertices

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