Uniform minimal full-access networks

Sridhar, M.A.; Raghavendra, C.S.

doi:10.1016/0743-7315(88)90004-4

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…6]. The concept is also known in connection to UPP graphs [51]. A recursive construction for a UPP graph using the line graph is not new.…”

Section: Constructions Of Upp Graphsmentioning

confidence: 99%

“…Three such constructions are known and they are presented in [26, 27, 51]. The first basic idea of exchanging the location of edges was presented by Sridhar and Raghavendra [51]. The idea was further developed by Goldfeld and Etzion [27] who proved that the number of nonisomorphic UPP graphs of order

n+1

is at least

{2}^{\theta (n)}

, where

\theta (n)

is the maximum size of an independent set (a set of vertices in which no two distinct vertices are connected by an edge) in

{G}_{n}

.…”

Section: Constructions Of Upp Graphsmentioning

confidence: 99%

“…One property of the constructions of exchanging edges in

{G}_{n}

[27, 51] is that all the alternating cycles have length 4 or 8. A complicated construction which yields also alternating cycles of length 12 is presented in [26], but other than that it is not known how to construct a UPP graph with longer alternating cycles.…”

Section: Constructions Of Upp Graphsmentioning

confidence: 99%

“…The main constructions to obtain nonisomorphic UPP graphs are based on modifications of the de Bruijn graph (by exchanging the location of some edges). Three such constructions are known and they are presented in [26,27,51]. The first basic idea of exchanging the location of edges was presented by Sridhar and Raghavendra [51].…”

Section: Constructions Based On Exchange Of Edges In De Bruijn Graphmentioning

confidence: 99%

“…As noted above, UPP graphs were defined and studied by Mendelsohn [46]. They were further studied in connection with interconnection networks and especially permutation networks [26,27,43,44,51,52], where they have found an application. Another application for cellular automata was found in [6].…”

Section: Introductionmentioning

confidence: 99%

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Graphs with the unique path property: Structure, cycles, factors, and constructions

Etzion

2023

Journal of Graph Theory

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In a directed graph with the unique path property, there is a unique directed path of a given length between any ordered pair of its vertices. Such graphs, their paths, cycles, factors, and other properties, were extensively studied during the years, mainly in the context of de Bruijn graphs, forming the most important family of such graphs. Such a graph is equivalent to a binary matrix for which , where is the all‐ones matrix. In this paper, some new results on the structural properties of such graphs, which are not necessarily de Bruijn graphs, are proved. We mainly concentrate on such graphs for which each vertex has in‐degree two and out‐degree two. The minimum number and the maximum number of vertices which are reached from a given vertex using all paths of length smaller than are considered. We provide an exact formula for the number of factors in these graphs and show that this number depends only on the number of alternating cycles in the graph, where in an alternating cycle every two consecutive edges have opposite directions in the cycle. Constructions of such graphs using several methods are presented. In particular, we present a new construction which provides a large set of such graphs with long alternating cycles. We define the integral graph and the derivative graph and implement these definitions on this family of graphs. Various properties from the point of view of the de Bruijn graph and the theory of nonsingular shift‐registers sequences are also considered. Finally, generalizations for graphs in which the in‐degree and the out‐degree of each vertex are greater than two are also discussed.

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“…6]. The concept is also known in connection to UPP graphs [51]. A recursive construction for a UPP graph using the line graph is not new.…”

Section: Constructions Of Upp Graphsmentioning

confidence: 99%