Showing papers on "Strongly regular graph published in 1981"
TL;DR: In this paper, it was shown that f(x ) is the expected eigenvalue distribution for every large randomly chosen labeled regular graph with degree v. It turns out that f (x)= v 4(v−1)−v 2 2 2π(v 2 −x 2 ) 0 for ∦ x ∦⩽2 v-1, unless √ x ∩⩾2 v −1, and f(X ) is a constant.
411 citations
TL;DR: The probability that a random labelled r -regular graph contains a given number of cycles of given length is investigated asymptotically and an asymPTotic formula results for the number of labelled r-regular graphs with a given girth.
173 citations
117 citations
01 Feb 1981
TL;DR: In this article, it was shown that there are cubic graphs of large girth without many independent vertices and that they can be constructed by probabilistic methods, and that their independence ratio is bounded away from 2.
Abstract: A set I of vertices of a graph G = (V, E) is said to be independent if no two vertices of I are joined by an edge of G. The maximal number of vertices in an independent set is the (vertex) independence number 1o(G) and /30(G)/IGI is the independence ratio of G. (Here I GI denotes the order of G, that is, the number of vertices of G. For the notation and general background, see [3].) The independence ratio is of great interest, not least because of its close relationship to the chromatic number. An immediate corollary of a classical theorem of Brooks [6] is that if a graph has maximal degree A = /\(G) and does not contain a KA+1, a complete graph of order A\ + 1, then its independence ratio is at least 1/A. The condition that G does not contain a KA+1 is clearly necessary; it is about the weakest condition expressing the fact that G is "not too dense". Lately several authors have given lower bounds for the independence ratio under other "sparseness" conditions (see [1], [2], [8]-[11]). The sparseness of a graph is usually measured by its girth, the minimal length of a cycle. Let i(/, g) be the infimum of the independence ratio of graphs with maximum degree A and girth at least g. In this notation Staton [9] proved that i(A, 4) > 5/(5A 1), and so in view of an example of Fajtlowicz [8] we have i(3, 4) = i(3, 5) = 5/14. Furthermore, concerning cubic graphs of large girth, Hopkins and Staton [10] showed that limgO, i(3, g) > 7/18. Our aim is to give upper bounds for i(A, g). In particular, we shall show that i(3, g) is bounded away from 2 which seemed the natural value for limg,. i(3, g) and, in fact, i(3, g) < 6/13 for every g. In order to do this one has to show that there are cubic graphs of large girth without many independent vertices. We shall not construct such graphs but show their existence by probabilistic methods. Probabilistic methods have been more and more in use since Erdos [7] tackled a similar problem over twenty years ago. However, the proof of our main result has become possible because of a very recent probabilistic model of regular graphs [5]. We state the theorem in its sharpest, though rather unattractive, form; later we deduce from it some more appealing and slightly weaker results.
87 citations
TL;DR: It is shown that for any two finite regular graphs G and H of the same degree, there exists a finite graph K that is simultaneously a covering both of G andH.
72 citations
TL;DR: It is shown that there exists no regular graph with excess e =1 and girth 2 r +1⩾5.
60 citations
TL;DR: In this paper, it was shown that none of these graphs have diameter greater than 5, except for some well-known exceptions, and hence they cannot be classified as generalised polygons.
Abstract: If a graph attains the bound (1.1) it is called a Moore graph if g is odd and a generalised polygon if g is even. A lot of work has been done on the classification of these graphs. Moore graphs have been studied by Hoffman and Singleton [8], Bannai and Ito [1], and Damerell [5]. Generalised polygons have been studied by Feit and Higman [7], Benson [3], and Singleton [9]. Biggs ([4]; Chapter 23) considered both types of graph as special cases of a more general family of distanceregular graphs. In this paper, we shall study this enlarged family of graphs. We shall prove that (apart from some well-known exceptions) none of them have diameter greater than 5.
18 citations
TL;DR: The inequality r < 14n + O(log n) + r the radius of G is proved and a similar theorem concerning any (2m −1)-connected graph G can be proved too.
14 citations
TL;DR: The coefficients of chromatic polynomials of certain connected graphs, relative to the tree basis, do not exhibit the strong logarithmic concavity property, and an infinite family of counterexamples to the conjecture that all regular graphs are chromatically unique are constructed.
12 citations
TL;DR: It is derived that a regular graph with diameter k and degree d is minimal d-connected when k ≧ 3 and Nk−d ≦ n < NK or k = 2 and n = N2−2.
Abstract: This paper discusses the relation between the diameter and the number of nodes of a regular graph; also discussed is the relation between a regular graph with minimum diameter and minimal d-connectivity from the viewpoint of graph theory for the construction of a communication network with high reliability and quality of communication.
First, it is shown that the upper bound of the number of nodes n for given diameter k is Nk−2 in almost all cases, where Nk is Moore's upper bound. Second, it is derived that a regular graph with diameter k and degree d is minimal d-connected when k ≧ 3 and Nk−d ≦ n < NK or k = 2 and n = N2−2.
10 citations
TL;DR: In this paper, some methods are given for constructing regular r-valent r-connected non-hamiltonian graphs; often the graphs are also non-r-edge-colorable.
Abstract: Some methods are given for constructing regular r-valent r-connected non-hamiltonian graphs; often the graphs are also non-r-edge-colorable. The extent of the class of such graphs constructible from these methods and previous methods is discussed.
TL;DR: The algebra of polynomials in the adjacency matrix is investigated and relate to every graph a family of orthogonal polynmials, which generalizes various results on distance regular graphs.
TL;DR: A new graph-acceptor model which can be considered as a canonical generalization of finite automata from strings to labelled (selector-) graphs is introduced.
Abstract: We introduce a new graph-acceptor model which can be considered as a canonical generalization of finite automata from strings to labelled (selector-) graphs.
TL;DR: It is shown that certain conditions assumed on a regular self-complementary graph are not sufficient for the graph to be strongly regular, answering in the negative a question posed by Kotzig in [1].
Abstract: It is shown that certain conditions assumed on a regular self-complementary graph are not sufficient for the graph to be strongly regular, answering in the negative a question posed by Kotzig in [1].
01 Jan 1981
TL;DR: A necessary and necessary condition for a computation graph to be regular and an algorithm to decide the regularity are presented and a necessary and sufficient condition for the change diagram to be realized by a live marked graph is given.
01 Jan 1981
TL;DR: In this article, the authors defined strong point-stability as the regular point-stable designs whose multigraphs are linear strongly regular, i.e., they have two or three eigenvalues.
Abstract: A design (incidence structure) D with incidence matrix A is called point stable, if AATJ=αJ (J the all-one-matrix, α ∈ ℕ). D is called strong, if D is a connected regular point stable design with two or three eigenvalues. The most important strong designs are the 2-designs, the partial geometric designs, (r,λ)-designs, regular point stable semi partial geometric designs, 2-PBIBD's and strongly regular graphs. The strong designs are characterized as the regular point stable designs whose multigraphs are linear strongly regular. The eigenvalues of strong designs may be expressed in geometrical terms. For any regular point stable design D we determine the “place” of the multigraph of D in the rank classification scheme. The point graph of a strong design with exactly two connection numbers is strongly regular.
TL;DR: In this paper, strongly regular graphs which are stable are characterized and shown to be strongly regular in the sense that they are stable in the presence of stable adjacency matrices.
Abstract: In this note we characterize strongly regular graphs which are stable.
TL;DR: A central result of this paper is that if G is regular and has girth g, then G is ( g − 1)-subgraph regular, which generalizes the familiar notion of regularity in graphs.