Showing papers on "Strongly regular graph published in 2000"
TL;DR: In this article, it was shown that if G is a connected graph of order n with minimum degree at least 2, then either Υt(G) ≤ 4n-7 or G e {C3, C5, C6, C10}.
Abstract: Let G = (V,E) be a graph. A set S ⊆ V is a total dominating set if every vertex of V is adjacent to some vertex in S. The total domination number of G, denoted by Υt(G), is the minimum cardinality of a total dominating set of G. We establish a property of minimum total dominating sets in graphs. If G is a connected graph of order n ≥ 3, then (see [3]) Υt(G) ≤ 2n-3. We show that if G is a connected graph of order n with minimum degree at least 2, then either Υt(G) ≤ 4n-7 or G e {C3, C5, C6, C10}. A characterization of those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G) e 2 and Υt(G) ≥ 4n-7 is obtained. We establish that if G is a connected graph of size q with minimum degree at least 2, then Υt(G) ≤(q + 2)-2. Connected graphs G of size q with minimum degree at least 2 satisfying Υt(G) > q-2 are characterized. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 2145, 2000
107 citations
TL;DR: In this article, it was shown that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most (n 2 + 1 + 5 ) + 5.
Abstract: The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ( n2)-1 Σ{x,y}‚V(G) dG(x, y), where V(G) denotes the vertex set of G and dG(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most $${n\over \delta + 1} + 5$$. We give improved bounds for K3-free graphs, C4-free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 113, 2000
58 citations
TL;DR: Chung and Garey as mentioned in this paper showed that the maximum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d is n-⌊d-2⌋ -O(1).
Abstract: Let fd (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n0 (D) vertices, f2(G) = n - D - 1 and f3(G) ≥ n - O(D3). For d ≥ 4, fd (G) depends strongly on the actual structure of G, not only on the maximum degree of G. We prove that the maximum of fd (G) over all connected graphs on n vertices is n-⌊d-2 ⌋ - O(1). As a byproduct, we show that for the n-cycle Cn, fd (Cn) = n-(2⌊d-2 ⌋ - 1) - O(1) for every d and n, improving earlier estimates of Chung and Garey in certain ranges. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 161–172, 2000
1991 Mathematics Subject classification: 05C12.
51 citations
TL;DR: In this article, it was shown that λ 2(G) ≪g(k-2) is the minimum cardinality of a set of edges in G such that G-S is disconnected and is of minimum degree at leasth.
Abstract: Letk andh be two integers, 0≤h
40 citations
TL;DR: G is thin if and only if G is thin, and the relationship between the Terwilliger algebras and their module structures of two graphs related in this way is investigated.
34 citations
TL;DR: It is shown that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular, and it is proved the nonexistence of distance- regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier.
Abstract: We find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Juri?i?, Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier .
28 citations
TL;DR: The representation number of the disjoint union of complete graphs to the existence of complete families of mutually orthogonal Latin squares is related and new results are obtained on representation numbers for several classes of graphs.
26 citations
Journal Article•
TL;DR: In this article, a 0.795 approximation algorithm for the Max-Bisection problem restricted to regular graphs was proposed, which implies an approximation ratio of 0.834.
Abstract: We design a 0.795 approximation algorithm for the Max-Bisection problem restricted to regular graphs. In the case of three regular graphs our results imply an approximation ratio of 0.834.
26 citations
Journal Article•
TL;DR: The aim of this paper is to draw together many of the known results about quasi-quadrics, as well as to provide some new geometric construction methods and theorems.
Abstract: In a projective space PG(n, q) a quasi-quadric is a set of points that has the same intersection numbers with respect to hyperplanes as a non degenerate quadric in that space Of course, non-degenerate quadrics themselves are examples of quasi-quadrics, but many other examples ex ist In the case that n is odd, quasi-quadrics have two sizes of inter sections with hyperplanes and so are two-character sets These sets are known to give rise to strongly regular graphs, two-weight codes, differ ence sets, SDP-designs, Reed-Muller codes and bent functions When n is even, quasi-quadrics have three sizes of intersection with respect to hyperplanes Certain of these may be used to construct antipodal dis tance regular covers of complete graphs The aim of this paper is to draw together many of the known results about quasi-quadrics, as well as to provide some new geometric construction methods and theorems
24 citations
TL;DR: A sharp upper bound of the Nordhaus–Gaddum type is obtained: ρ(G)+ρ(G c )⩽ 2− 1 k − 1 k n(n−1) , where k and k are the chromatic numbers of G and G c , respectively.
24 citations
TL;DR: Using a different and more efficient method, it is discovered that there are precisely $28$ non-isomorphic strongly regular $(40,12,2,4) graphs.
Abstract: In a previous paper it was established that there are at least $27$ non-isomorphic strongly regular $(40,12,2,4)$ graphs. Using a different and more efficient method we have re-investigated these graphs and have now been able to determine them all , and so complete the classification. We have discovered that there are precisely $28$ non-isomorphic $(40,12,2,4)$ strongly regular graphs. The one that was not found in the previous investigation is characterised uniquely by the fact that every neighbour graph is triangle-free.
TL;DR: It is shown that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a HadamARD matrix.
Abstract: A graph is $3$-e.c. if for every $3$-element subset $S$ of the vertices, and for every subset $T$ of $S$, there is a vertex not in $S$ which is joined to every vertex in $T$ and to no vertex in $S\setminus T$. Although almost all graphs are $3$-e.c., the only known examples of strongly regular $3$-e.c. graphs are Paley graphs with at least $29$ vertices. We construct a new infinite family of $3$-e.c. graphs, based on certain Hadamard matrices, that are strongly regular but not Paley graphs. Specifically, we show that Bush-type Hadamard matrices of order $16n^2$ give rise to strongly regular $3$-e.c. graphs, for each odd $n$ for which $4n$ is the order of a Hadamard matrix.
TL;DR: It is proved that G has an [ a, b ]-factor if the minimum degree δ ( G )⩾ a, n ⩾2( a + b )( a+ b −1)/ b and | N G ( x )∪N G ( y )|⩽ an /( a - b ) for any two non-adjacent vertices x and y of G.
TL;DR: It is proved that if q = 2 the corresponding strongly regular graphs are switching equivalent and that they contain subconstituent graphs that are point graphs of partial geometries.
Abstract: Two-weight codes and projective sets having two intersection sizes with hyperplanes are equivalent objects and they define strongly regular graphs We construct projective sets in \PG(2m-1,q) that have the same intersection numbers with hyperplanes as the hyperbolic quadric \Q^{+}(2m-1,q) We investigate these sets; we prove that if q=2 the corresponding strongly regular graphs are switching equivalent and that they contain subconstituents that are point graphs of partial geometries If m=4 the partial geometries have parameters s=7, t=8, \alpha = 4 and some of them are embeddable in Steiner systems \S(2,8,120)
TL;DR: It is shown that ρ(K 2 × C 2m )=2, ρ (K n ×C 2m)=1 for n=3,4,5,7, and ρ-(Kn×C2m)=0 for most cases otherwise.
TL;DR: It is shown that there exists a constant c such that for any integers r,k?2 and for any coloring of the edges of a complete graph with r colors, its vertices can be partitioned into at most rc(rlogr+k) connected monochromatic k-regular subgraphs and vertices.
TL;DR: A characterization of strongly regular graphs with diameter d in terms of the eigenvalues, the sum of the multiplicities corresponding to the Eigenvalues with (non-zero) even subindex, and the harmonic mean of the degrees of the distance-$d$ graph.
Abstract: A graph $\Gamma$ with diameter $d$ is strongly distance-regular if $\Gamma$ is distance-regular and its distance-$d$ graph $\Gamma _d$ is strongly regular. The known examples are all the connected strongly regular graphs (i.e. $d=2$), all the antipodal distance-regular graphs, and some distance-regular graphs with diameter $d=3$. The main result in this paper is a characterization of these graphs (among regular graphs with $d$ distinct eigenvalues), in terms of the eigenvalues, the sum of the multiplicities corresponding to the eigenvalues with (non-zero) even subindex, and the harmonic mean of the degrees of the distance-$d$ graph.
TL;DR: In this article, a complete classification of distance-regular graphs of valency 6 and a1 = 1 is given, where a 1 is the number of vertices in the graph.
Abstract: We give a complete classification of distance-regular graphs of valency 6 and a1 = 1.
TL;DR: In this paper, it was shown that maximal arcs in symplectic translation planes can be obtained from certain m-systems of finite symplectic polar spaces, and many new examples of maximal arcs are then constructed.
Abstract: Shult and Thas have shown in [13] that m-systems of certain finite classical polar spaces give rise to strongly regular graphs and two weight codes. The main result of this paper is to show that maximal arcs in symplectic translation planes may be obtained from certain m-systems of finite symplectic polar spaces. Many new examples of maximal arcs are then constructed. Examples of m-systems are also constructed in Q(-)(2n + 1,q) and W2n+1(q). A method different from that of Shult and Thas is used to construct strongly regular graphs using "differences" of m-systems.
TL;DR: In this article, the complete computer search for a strongly regular graph with parameters (36,15,6,6) and chromatic number six was conducted and the result is that no such graph exists.
Abstract: We report on the complete computer search for a strongly regular graph with parameters (36,15,6,6) and chromatic number six. The result is that no such graph exists.
TL;DR: It is proved that the Heawood graph is an instance of a graph ful lling this theorem minimally, i.e. in which every one-factor belongs to precisely one one-Factorization.
TL;DR: It is shown that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined, which is within a constant factor of the optimum.
Abstract: Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is NP-complete, although it is in P if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.
TL;DR: This work considers L as a kernel on the finite space V(?), in the context of the Potential Theory, and proves that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(?).
Abstract: We aim here to introduce a new point of view of the Laplacian of a graph, ?. With this purpose in mind, we consider L as a kernel on the finite space V(?), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(?). If ? is a proper set of a suitable host graph, then the equilibrium problem for ? can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of ?. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs.
TL;DR: A general bound for the irregularity strength of regular compound graphs is proved and exact results for some infinite families of graphs are derived.
TL;DR: In this article, it was shown that the conjecture is true for every integer k = 0, 1 and for any integer k ≥ 0, it is proven that k + 1 edge disjoint Hamilton cycles exist.
Abstract: Let G be a graph of order n and k≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)-connected graph G,d G (u,v) = 2 implies that max{d(u;G), d(v;G)}≥ (n-2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k≥0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 820, 2000
Dissertation•
01 Jan 2000
TL;DR: In this article, the authors present a methodology for the systematic generation of strongly regular graphs by fusing classes in large association schemes, which includes an algorithm for finding the eigenvalue matrix of an association scheme; an algorithm that determines the lattice of matrices determined by subgroups of the automorphism group of an associative scheme; and an integer programming formulation that allows us to determine quickly whether a matrix (obtained through fusing the columns of the eigvalue matrix according to some group) will yield strongly regular graph with a given parameter set.
Abstract: This thesis presents a methodology for the systematic generation of strongly regular graphs by fusing classes in large association schemes; this methodology includes: (1) an algorithm for finding the eigenvalue matrix of an association scheme; (2) an algorithm for determining the lattice of matrices determined by subgroups of the automorphism group of an association scheme; (3) an integer programming formulation that allows us to determine quickly whether a matrix (obtained by fusing the columns of the eigenvalue matrix of an association scheme according to some group) will yield strongly regular graphs with a given parameter set; (4) an exhaustive search algorithm that, given the parameters of a graph and a matrix, will find all the combinations of columns of the matrix that correspond to strongly regular graphs with the given parameters.
Using the methodology presented here, we have settled the existence problem for strongly regular graphs with the following parameters:
srg(625, 156, 29, 42); srg(1024, 330, 98, 110); srg(729, 308, 127, 132); srg(1024, 363, 122, 132); srg(729, 336, 153, 156); srg(1024, 396, 148, 156); srg(1024, 231, 38, 56); srg(1024, 429, 176, 182); srg(1024, 297, 76, 90); srg(1024, 462, 206, 210).
We have also found new graphs for many parameter sets for which some graphs were already known, as well as graphs which were themselves known.
Many of the graphs found are pseudo-geometric, i.e., their parameters correspond to those of the point graph of some hypothetical finite geometry. Some of the graphs are actually geometric, and it is possible to recover the geometry from the maximum cliques of the graph.
The graphs presented here are all of cyclotomic type; therefore they all can be expressed as two-intersection sets in suitable projective geometries, and they all determine two-weight codes, but the methodology is not restricted to this kind of graphs. Several results concerning two-intersection sets are derived.
We have found six interesting non-isomorphic strongly regular graphs with parameters (625, 156, 29, 42). If we consider the corresponding two-intersection sets in PG(3, 5), the affine lines determined by each of these sets (which are 5-cliques in the graph) form a geometric structure that resembles a semipartial geometry: given any anti-flag (p, L), the number of points in L which are collinear with p is either 0, 1 or 2 (i.e., this number takes one of three values, while in the case of partial geometries it takes only one value and in the case of semipartial geometries it takes one of two values).
Another interesting property of these graphs is that it is possible to factorize the complete graph K625 into copies of some of these graphs; we exhibit three such factorizations, each of them cyclic.
We have also developed algorithms to perform computations in a projective geometry; the idea in which these algorithms are based is to represent each d-dimensional subspace of PG(n, q) by a canonical basis (set of d+1 points that span the subspace); this canonical basis can be obtained—from any other basis—by a process based in Gaussian elimination.
TL;DR: In this paper, sharp upper bounds on Γ − of regular graphs are found and are found to answer an open problem proposed by Henning and Slater.
TL;DR: The main purpose of this paper is to give a complete solution to the problem of determining which Cayley graphs are graphical regular representations of the corresponding groups for the class of metacyclic p -groups where p is a prime.
Abstract: A Cayley graph ?=Cay(G, S) is called a graphical regular representation of the group G if Aut?=G. One long-standing open problem about Cayley graphs is to determine which Cayley graphs are graphical regular representations of the corresponding groups. A simple necessary condition for ? to be a graphical regular representation of G isAut (G, S) = 1, where Aut(G, S) = { ??Aut(G) |S?=S }. C. Godsil in (Europ. J. Combinatorics, 4 (1983)) proposed to characterize graphical regular representations of groups G in terms ofAut (G, S); that is, for a given class of groups G, find the conditions under whichCay (G, S) is a graphical regular representation of G if and only ifAut (G, S) = 1. The main purpose of this paper is to give a complete solution to this problem for the class of metacyclic p -groups where p is a prime.
Journal Article•
TL;DR: In this article, the domination number of a connected graph of order p is characterized, and the main results are as follows: (1) when p is even, γ(G) = p/2 if and only if either G C4 or G is the crown of a given connected graph with p 2 vertices.
Abstract: Let G be a connected graph of order p, and let γ7(G) denote the domination number of G. Clearly, γ(G) ≤[p/2]. The aim of this paper is to characterize the graphs G that reaches this upper bound. The main results are as follows: (1) when p is even, γ(G) = p/2 if and only if either G C4 or G is the crown of a connected graph with p/2 vertices; (2) when p is odd, γ(G) = (p-1)/2 if and only if every spanning tree of G is one of the two classes of trees shown in Theorem 3.1.
TL;DR: The fundamental properties of the three subgraphs H, H′ and H ∗ of the self-complementary graph G are considered in detail at first and these results will be used to study certain Ramsey number problems.