Large Cayley Graphs and Digraphs with Small Degree and Diameter

Hafner, Paul R.

doi:10.1007/978-94-017-1108-1_21

Cited by 20 publications

(13 citation statements)

References 26 publications

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“…[4]) that this bound is attained only for d = 1, 2, 3, 7, and possibly 57 (the corresponding graphs are known as Moore graphs of diameter two). Results of [1] show that f (d) ≤ d 2 − 1 for the remaining values of d. So far we do not have satisfactory lower bounds (see [3] for a table of bounds on f (d) for d ≤ 15); the general current record construction comes from forgetting arrows in line digraphs of complete digraphs, giving f (d) ≥ (d/2) (d + 2)/2 for each d.…”

Section: Introductionmentioning

confidence: 96%

Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface

Knor

?ir�?

1997

J. Graph Theory

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It is known that for each d there exists a graph of diameter two and maximum degree d which has at least ⌈(d/2)⌉ ⌈(d + 2)/2⌉ vertices. In contrast with this, we prove that for every surface S there is a constant ds such that each graph of diameter two and maximum degree d ≥ ds, which is embeddable in S, has at most ⌊(3/2)d⌋ + 1 vertices. Moreover, this upper bound is best possible, and we show that extremal graphs can be found among surface triangulations. © 1997 John Wiley & Sons, Inc.

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Section: Introductionmentioning

confidence: 96%

Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface

Knor

?ir�?

1997

J. Graph Theory

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“…K As remarked in the Introduction, the above construction gives, for d=7, the Hoffman Singleton graph [15]. For d=11 and 13 we obtain graphs of diameter 2 on 98 and 162 vertices, which are larger than any of the previously known graphs of diameter 2 and degrees 11 and 13 (compared with the values 94 and 136 contained in the tables of currently largest graphs of given degree and diameter in [8] or [14]). Especially, the new vertextransitive graph of order 162 is of great interest as it misses the Moore bound only by 8.…”

Section: Let H Q =Gmentioning

confidence: 95%

“…Even more importantly, for d=11 and d=13 we obtain new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively (the largest previously known orders were 94 and 136, cf. [8,14]). …”

Section: Introductionmentioning

confidence: 97%

A Note on Large Graphs of Diameter Two and Given Maximum Degree

McKay

Miller

Širáň‬

1998

Journal of Combinatorial Theory, Series B

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Let vt(d, 2) be the largest order of a vertex-transitive graph of degree d and diameter 2. It is known that vt(d, 2)=d 2 +1 for d=1, 2, 3, and 7; for the remaining values of d we have vt(d, 2) d 2 &1. The only known general lower bound on vt(d, 2), valid for all d, seems to be vt(d, 2) w(d+2)Â2x W(d+2)Â2X. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d, 2) (8Â9)(d+ 2 for all d of the form d=(3q&1)Â2, where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d=7 we obtain as a special case the Hoffman Singleton graph, and for d=11 and d=13 we have new largest graphs of diameter 2, and degree d on 98 and 162 vertices, respectively.

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“…For instance, they often make good models for interconnection networks; see papers by Hafner [21], Heydemann et al [22,23], Lakshmivarahan et al [28], and other references therein. They also have applications to the study of finitely-generated groups and permutation groups, as in the work of Trofimov [38].…”

Section: Introductionmentioning

confidence: 98%

On symmetries of Cayley graphs and the graphs underlying regular maps

Conder

2009

Journal of Algebra

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By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of valency 3 t + 1 that are 7-arc-transitive, for all t > 0. The same approach can be taken in considering the graphs underlying regular or orientably-regular maps, leading to classifications of all such maps having a 1-, 4-or 5-arc-regular 3-valent underlying graph (in answer to questions by Cheryl Praeger and Sanming Zhou).

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Large Cayley Graphs and Digraphs with Small Degree and Diameter

Abstract: Abstract. W e review the status of the Degree/Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.

Cited by 20 publications

References 26 publications

Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface

Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface

A Note on Large Graphs of Diameter Two and Given Maximum Degree

On symmetries of Cayley graphs and the graphs underlying regular maps

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