Showing papers on "Strongly regular graph published in 1993"
TL;DR: A graph may be regarded as an electrical network in which each edge has unit resistance and explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case are obtained.
Abstract: A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case. A well-known link with random walks motivates a conjecture about the maximum effective resistance. Arguments are given that point to the truth of the conjecture for all known distance-regular graphs.
40 citations
TL;DR: In this paper, a strongly regular semi-Cayley graph corresponds to a triple of subsets (C, D, D′) of a group G, which is called a partial difference triple.
Abstract: It is known that a strongly regular semi-Cayley graph (with respect to a group G) corresponds to a triple of subsets (C, D, D′) of G. Such a triple (C, D, D′) is called a partial difference triple. First, we study the case when D ∪ D′ is contained in a proper normal subgroup of G. We basically determine all possible partial difference triples in this case. In fact, when \vert G\vert
eq 8 nor 25, all partial difference triples come from a certain family of partial difference triples. Second, we investigate partial difference triples over cyclic group. We find a few nontrivial examples of strongly regular semi-Cayley graphs when vGv is even. This gives a negative answer to a problem raised by de Resmini and Jungnickel. Furthermore, we determine all possible parameters when G is cyclic. Last, as an application of the theory of partial difference triples, we prove some results concerned with strongly regular Cayley graphs.
38 citations
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TL;DR: In this paper, the authors determined all graphs with the spectrum of a distance-regular graph with at most 30 vertices, except possibly for the Taylor graph on 28 vertices.
Abstract: We determine all graphs with the spectrum of a distance-regular graph with at most 30 vertices (except possibly for the Taylor graph on 28 vertices)
29 citations
TL;DR: The graph obtained by removing any k − m edges of G, has an m-factor, which indicates that the graph with an even number of vertices has an M-factor.
26 citations
TL;DR: This paper shows that a graph G contains such a cycle provided it has any of the following three properties: G has minimum degree at least 2 and at most two vertices of degree 2, G is not 3-colorable, and G is a subdivision of a graph of order p ⩾5 with at least 3 p -5 edges.
25 citations
TL;DR: The main aim of this paper is not to present new results but to give a short survey on some relations between graphs and configurations.
25 citations
TL;DR: In this article, the use of regular graph colouring as an equivalent simple definition for regular equivalence is extended from graphs to digraphs and networks, and new concepts of regular equivalences for edges and hypergraphs are presented using the new terminology.
24 citations
TL;DR: It is proved that, if G is an (n−3)- or ( n−4)- regular graph of order n, the strength of G is 3 (except if G=K3,3); and it is conjecture that the irregularity strength of an r-regular graph ofOrder n⩽2r is 3, except ifG is Kl,l with l odd.
22 citations
01 Jan 1993
18 citations
TL;DR: It is shown that every 3-connected claw-free graphs having at most 5?
TL;DR: In this paper, it was shown that a claw-free graph G is hamiltonian if and only if G+uv is h amiltonian, where u,v is a K4-pair.
TL;DR: This work determines the edge-toughness T1(G) of a graph G where the minimum is taken over every edge-cutset X that separates G into ω (G - X) components and gives the arboricity of these graphs.
Abstract: The edge-toughness T1(G) of a graph G is defined as
where the minimum is taken over every edge-cutset X that separates G into ω (G - X) components. We determine this quantity for some special classes of graphs that also gives the arboricity of these graphs. We also give a simpler proof to the following result of Peng et al.: For any positive integers r, s satisfying r/2 < s ≤ r, there exists an infinite family of graphs such that for each graph G in the family, λ(G) = r (where λ(G) is the edge-connectivity of G) T1(G) = s, and G can be factored into s spanning trees. © 1993 John Wiley & Sons, Inc.
19 Jul 1993
TL;DR: This work gives several sufficient conditions for 4-regular graph to have a 3-regular subgraph.
Abstract: We are interested in the following problem: when would a 4-regular graph (with multiple edges) have a 3-regular subgraph. We give several sufficient conditions for 4-regular graph to have a 3-regular subgraph.
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Dissertation•
01 Jan 1993
TL;DR: In this article, it was shown that the strongly regular graphs with p(m,n,k) ≥ 4n-1 (2(n+k) + ½ (3 = (-1 n+k+1+1} + 1/3 l1/3 ) for m = n = 1, graphs with this property (k,t + 1, n 2 3 ) for rn 2 3 -n 2 -n 3 -1, are the graphs with properties P(m.n, n,k).
Abstract: A graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices there are at least k other vertices, each of which is adjacent to the first m vertices but not adjacent to any of the latter n vertices. The class of graphs having property P(m.n,k) is denoted by ζ(m,n,k). The problem that arises is that of characterizing the class ζ(m,n,k). One particularly interesting problem that arises concerns the functionP(m,n,k) = min{υ(G) : G є ζ(m,n,k) }.In Chapter 2, we establish some important properties of graphs in the class ζ(m,n,k) and a lower bound on p(m,n,k). In particular, we prove thatp(n,n,k) ≥ 4n-1 (2(n+k) + ½ (3 = (-1)n+k+1} + 1/3 l 1/3One of the results in Chapter 2 is that almost all graphs have property P(m,n,k). However, few members of ζ(m,n,k) have been exhibited. In Chapter 3. we construct classes of graphs having property P(l,n,k) . These classes include the cubes, "generalized" Petersen graphs and "generalized" Hoffman-Singleton graphs.An important graph in the study of the class ζ(m,n,k) is the Paley graph Gq defined as follows. Let q = l(mod 4) be a prime power. The vertices of Gq are the elements of the finite field IFq. Two vertices a and b are joined by an edge if and only if their difference is a quadratic residue, that is a - b = y2 for some y є IFq. In chapter 4, we prove that for a prime p = l(mod 4), all sufficiently large Paley graphs GP satisfy property P(m.n,k). This is established by making use of results from prime number theory.In Chapter 5 , we establish, by making use of results from finite fields, the adjacency properties of Paley graphs of order q = pd , with p a prime.For directed graphs, there is an analogue of the above adjacency property concerning tournaments. A tournament Tq of order q is said to have property Q(n,k) if every subset of n vertices of Tq is dominated (if there is an arc directed from a vertex u to a vertex v, we say that u dominates v and that v is dominated by u) by at least k other vertices.Let q = 3(mod 4) is a prime power. The Paley tournament Dq is defined as follows. The vertices of Dq are the elements of the finite field IFq. Vertex a is ioined to vertex b by an arc if and only if a - b is a quadratic residue in Fq. In Chapter 6, we prove that the Paley tournament Dq has property Q(n,k) wheneverq > {(n - 3)2n-1 + Z}G + kZn - 1. A graph G is said to have property P*(rn,n,k) if for any set of rn + n distinct vertices of G there are exactly k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The class of graphs having property P*(m.n,k) is denoted by S*(m,n.k). The class S*(m,n,k) has been studied when one of m or n is zero. In Chapter 7, we show that, for m = n = 1, graphs with this property (k + t)' + 1, are the strongly regular graphs with parameters ( k + t,t - 1,t) for some positive integer t. For rn 2 1, n 2 1, and m + n 2 3, we show that, there is no graph having property P*(m.n,k), for any positive integer k. The first Chapter of this thesis provides the…
TL;DR: Since a chordal graph G can be expressed as the intersection graph of some subtrees of a tree, its rank rk(G) is defined to be the least extent of such a representation tree.
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TL;DR: It is proved that in a graph of maximum degree k every m-ASP has size m?k(k + 1)2 + 1, and conjecture that this bound can be improved to m?(p + pq), and verify this conjecture when the graph satisfies some additional assumptions.
TL;DR: It is shown that bi ⩽ l whenever ci = b1 whenever b0, b1, ..., bd−1 is a distance regular graph.
TL;DR: The only spin models associated with symmetric conference graphs with n ≥ 5 vertices are the pentagon and the lattice graph L2(3) with 9 vertices.
Abstract: We give a short and alternative proof of a theorem of F Jaeger that except for Potts models attached to the complete graphs, the only spin models associated with symmetric conference graphs with n ≥ 5 vertices are the pentagon and the lattice graph L2(3) with 9 vertices The proof avoids Jaeger's use of the classification of strongly regular graphs having strongly regular subconstituents due to P J Cameron, J M Goethals, and J J Seidel