Showing papers on "Strongly regular graph published in 1979"
105 citations
TL;DR: The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G as discussed by the authors, and all graphs with G ≥ −2 are characterized.
56 citations
TL;DR: It is shown that no regular graphs of girth 10 and valency 3 having fewer than 70 vertices exist.
36 citations
TL;DR: A (0,@l)-graph (@l>=2) is a connected graph in which any two vertices have @l common neighbours or none at all, and such a graph is regular when the diameter is at least four.
34 citations
TL;DR: It is proved that every finite simple 4-regular graph G contains a 3-regular subgraph if and only if the cyclic edge connectivity λc(G) of G is at least 10.
Abstract: Berge conjectured that every finite simple 4-regular graph G contains a 3-regular subgraph. We prove that this conjecture is true if the cyclic edge connectivity λc(G) of G is at least 10. Also we prove that if G is a smallest counterexample, then λc(G) is either 6 or 8.
28 citations
TL;DR: The two basic theorems proved here are (1) every graph is realizable and for any graph G, Spec G contains all but a finite set of integers.
Abstract: An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.
25 citations
TL;DR: All planar connected graphs regular of degree four can be generated from the graph of the octahedron, using four operations.
Abstract: All planar connected graphs regular of degree four can be generated from the graph of the octahedron, using four operations.
24 citations
TL;DR: It is shown that neither (*) nor (**) is a necessary condition for the existence of a 1-factorization of a Cartesian product of regular graphs.
Abstract: Let G1, G2…, Gn be regular graphs and H be the Cartesian product of these graphs (H = G1 × G2 × … × Gn). The following will be proved: If the set {G1, G2…, Gn} has at leat one of the following properties: (*) for at leat one i ϵ {1, 2,…, n}, there exists a 1-factorization of Gi or (**) there exists at least two numbers i and j such that 1 ≤ i < j ≤ n and both the Graphs Gi and Gj contain at least one 1-factor, then there exists a 1-factorization of H. Further results: Let F be a cycle of length greater than three and let G be an arbitrary cubic graph. Then there exists a 1-factorization of the 5-regular graph H = F × G. The last result shows that neither (*) nor (**) is a necessary condition for the existence of a 1-factorization of a Cartesian product of regular graphs.
23 citations
01 Feb 1979
TL;DR: In this article, it was shown that for any connected cubic graph G with 2n points, the number of automorphisms of G divides 3n 2 n. This is a special case of a result which is proved for connected regular graphs in general.
Abstract: For any connected cubic graph G with 2n points, the number of automorphisms of G divides 3n2n. This is a special case of a result which is proved for connected regular graphs in general. The result is shown to be best possible for infinitely many n in the cubic case. An r-regular graph is a graph G in which each point is of degree r; if r = 3 we say G is cubic. We have shown elsewhere [6] that the number of labelled connected cubic graphs with 2n points is divisible by (2n)!/(3n2'), as are the numbers of labelled 2-connected and 3-connected cubic graphs. This was done by obtaining recurrence relations for the numbers involved. Our main concern is to verify this result directly by showing that the number of ways to label any connected cubic graph with 2n points is divisible by (2n)!/(3n2'). Let s denote the order of the automorphism group F(G) of the graph G. If G is cubic and has 2n points, it follows [5] that the number of ways to label G is just (2n)!/s. Thus, our aim is fulfilled if we show that s divides 3n2n. The following theorem is a generalisation of this result. All basic graph theoretic notation not defined here can be found in [4]. THEOREM 1. Let G be a connected r-regular graph with p points where r > 0. Then the number s of automorphisms of G divides
8 citations
TL;DR: In this paper, the adjacency matrix of an undirected graph is the n x n matrix A = A(r), over the complex number field, whose entries aij are given by a,j = 1 if vi and vi are adjacent, = 0 otherwise.
TL;DR: It is proved that each connected component of G is either a complete graph, an arc or a polygon and explicit representation of μ based on the structure of G can be represented as a sum.
Abstract: Let S be a set and let μ be a mapping which assigns to each finite complete probability distribution P on S a real value μ ( P ). We refer to such a mapping μ as a measure of information in a broad sense. To each measure μ there is associated a simple graph G on S which indicates the extent μ can be localized. It is proved that each connected component of G is either a complete graph, an arc or a polygon. Explicit representation of μ based on the structure of G is given. In particular if G is the complete graph on S , then μ can be represented as a sum.
TL;DR: In this paper, the authors studied the problem of determining the values of p and k for which there is a line graph on p points that is regular of degree k. This question was suggested by F. T. Boesch and inspired by a recent survey paper by Hakimi and Schmeichel.
Abstract: The purpose of this note is to determine the values of p and k for which there is a line graph on p points that is regular of degree k. This question arises from the study of a more general question; namely, when is a given degree sequence realizable as a line graph? This problem was suggested by F. T. Boesch and inspired by a recent survey paper by Hakimi and Schmeichel [ 13 in which a “degree” is associated with each edge of a graph. The solution for the particular case of regular graphs is obtained by combining (1) known conditions for the existence of k-regular graphs and of (c, b)-biregular bipartite graphs, and (2) the fact that L(G) is regular if and only if G is regular or bipartite biregular. The solution is summarized in the iollowing theorem. Terminology and notation will be standard, as in Harary [2].
TL;DR: A graph G is said to be regular if and only if every vertex of G has the same vertex degree and three distinct vertices are said to form a triangle if they are pairwise adjacent.