Some large graphs with given degree and diameter

Alegre, I.; Fiol, M. A.; Yebra, J. L. A.

doi:10.1002/jgt.3190100211

Cited by 21 publications

(41 citation statements)

References 3 publications

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“…Let f be a rectagraph with valency k. Then f has at most 2k vertices, with equality if and only if f is the k-cube. Therefore, k;+l(y)(i + I) ...,; N; ...,; k;(y)(ki); by induction we find k;(y) ...,; (7). Fix y E f, and let N; be the number of edges a/3 with a E f;(y) and /3 Ef;+l(Y)· By Proposition 1.…”

Section: 2 a Rectagraph Is Regular;mentioning

confidence: 88%

“…These families of examples overlap for small parameters: (v, a, I) = (7,2,3) is found under both b) and e), (v,a,/) = (13,3,3) under both d) and e) and (v,a,/) = (21,4,3) under both c) and e). These families of examples overlap for small parameters: (v, a, I) = (7,2,3) is found under both b) and e), (v,a,/) = (13,3,3) under both d) and e) and (v,a,/) = (21,4,3) under both c) and e).…”

Section: Problem Prove That No Other Examples Existmentioning

confidence: 93%

“…~+3-p., (7) and, with b2 = min{b2(a,p) I d(a,p) = 2}, every 2-claw is in at most max{O,k -~ -3 + p. -b2} quadrangles. ~+3-p., (7) and, with b2 = min{b2(a,p) I d(a,p) = 2}, every 2-claw is in at most max{O,k -~ -3 + p. -b2} quadrangles.…”

Section: (I)mentioning

confidence: 99%

“…2 1 . 7. ;;;; d}; For more information on (mostly 3-c1ass) pseudocyclic association schemes, see §12.…”

Section: B Pseudocyclic Association Schemesmentioning

confidence: 99%

“…is an integer, and by (7), the basis b is an algebraic integer. We now consider the cases ( =1= ° and ( = ° separately.…”

Section: Graphs With Classical Parametersmentioning

confidence: 99%

See 4 more Smart Citations

Distance-Regular Graphs

1989

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Section: 2 a Rectagraph Is Regular;mentioning

confidence: 88%

Section: Problem Prove That No Other Examples Existmentioning

confidence: 93%

Section: (I)mentioning

confidence: 99%

“…2 1 . 7. ;;;; d}; For more information on (mostly 3-c1ass) pseudocyclic association schemes, see §12.…”

Section: B Pseudocyclic Association Schemesmentioning

confidence: 99%

“…is an integer, and by (7), the basis b is an algebraic integer. We now consider the cases ( =1= ° and ( = ° separately.…”

Section: Graphs With Classical Parametersmentioning

confidence: 99%

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Distance-Regular Graphs

1989

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Maximally connected digraphs

Fàbrega

Fiol

1989

Journal of Graph Theory

Self Cite

131

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This paper introduces a new parameter / = / ( G ) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let K, A, 6, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that A = 6 if D s 21 and K = 6 if D I 21 -1. Analogous results involving upper bounds for K and A are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-A and super-rc are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized. DIGRAPHS AND LINE DIGRAPHSThis paper concentrates on the connectivity of digraphs. More precisely, we study the relation between the connectivity, the diameter, and a new parameter that, in the case of graphs, is closely related with the girth. The results obtained have some interesting corollaries. For example, it is shown that iterated line digraphs are maximally connected if the iteration order is large enough.Let us first recall the notation used throughout the paper. Let G = ( K A ) denote a digraph with (finite) set of vertices V = V(G) and set of arcs A = A(G), which are ordered pairs of (not necessarily different) vertices of V. So, loops are allowed but parallel arcs are not. If e = (x, y ) E A, we say that x is adjacent to y and that y is ad'ucenrfrom x . Let r-(x) and I'+ (x) denote respectively the sets of vertices adjacent to and from x , i.e., the sets of in-neighbors and out-neighbors of x . Their cardinalities are the in-degree of x , K ( x ) = lI'-(x)l, and the out-degree of x , 6'(x) = Ir'(x)[. The minimum degree of G, 6 = 6(G), is the minimum over all the in-degrees and out-degrees of the vertices of G.

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Designing networks: A mixed‐integer linear optimization approach

et al. 2016

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Designing networks with specified collective properties is useful in a variety of application areas, enabling the study of how given properties affect the behavior of network models, the downscaling of empirical networks to workable sizes, and the analysis of network evolution. Despite the importance of the task, there currently exists a gap in our ability to systematically generate networks that adhere to theoretical guarantees for the given property specifications. In this paper, we propose the use of Mixed-Integer Linear Optimization modeling and solution methodologies to address this Network Generation Problem. We present a number of useful modeling techniques and apply them to mathematically express and constrain network properties in the context of an optimization formulation. We then develop complete formulations for the generation of networks that attain specified levels of connectivity, spread, assortativity and robustness, and we illustrate these via a number of computational case studies.

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Some large graphs with given degree and diameter

Cited by 21 publications

References 3 publications

Distance-Regular Graphs

Distance-Regular Graphs

Maximally connected digraphs

Designing networks: A mixed‐integer linear optimization approach

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