[Show abstract][Hide abstract] ABSTRACT: It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist(see  or ). Furthermore, for degree 2, it is shown that for k 3 there are nodigraphs of order `close" to, i.e., one less than, Moore bound . In this paper, weshall consider digraphs of diameter k, degree 3 and number of vertices one less thanMoore bound. We give a necessary condition for the existence of such digraphs and,using this condition, we deduce that such digraphs do not exist for infinitely ...
[Show abstract][Hide abstract] ABSTRACT: A simple method is proposed to find the orthogonal projection of a given point to the solution set of a system of linear equations. This is also a direct method for solving systems of linear equations. The output of the method is either the projection or inconsistency of the system. Moreover, in the process also linearly dependent equations are recognized. This paper is constrained for giving theoretical foundations, computational complexity and some numerical experiments with dense matrices although the method allows to employ sparsity. The raw method could not compete with best software packages in solving linear equations for general matrices, but it was competitive in finding projections for matrices with small number of rows relative to the number of columns.
Linear Algebra and Its Applications - LINEAR ALGEBRA APPL. 01/2007; 422(2):455-470.
[Show abstract][Hide abstract] ABSTRACT: Bounds on the sum and product of the chromatic numbers of n factors of a complete graph of order p are shown to exist. The well-known theorem of Nordhaus and Gaddum solves the problem for n = 2. Strict lower and some upper bounds for any n and strict upper bounds for n = 3 are given. In particular, the sum of the chromatic numbers of three factors is between 3p1/3 and p + 3 and the product is between p and [(p + 3)/3]3.
Journal of Graph Theory 10/2006; 2(1):9 - 17. · 0.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The nonexistence of digraphs with order equal to the Moore bound Mddk = 1 + d+: : : +d k for dd k > 1 has lead to the study of the problem of the existence oalmost' Moore digraphs, namely digraphs with order close to the Moore bound. In 1], it was shown that almost Moore digraphs of order Mddk ; 1, degree d, diameter k (dd k 3) contain either no cycle of length k or exactly one such cycle. In this paper we shall derive some further necessary conditions for the existence of almost Moore digraphs for degree and diameter greater than 1. As a consequence, for diameter 2 and degree d, 2 d 12, we show that there are no almost Moore digraphs of order Mdd2 ; 1 with one vertex in a C2 except the digraphs with every vertex in C2 .
Ars Combinatoria -Waterloo then Winnipeg- 01/2000; 56. · 0.28 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem,whichis to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and di...
[Show abstract][Hide abstract] ABSTRACT: The Moore bound for a diregular digraph of degree d and diameter k is M d;k =1 + d + : : : + dk. It is known that digraphs of order M d;k do not exist for d ? 1and k ? 1 ( or ). In this paper we study digraphs of degree d, diameter kand order M d;k \Gamma 1, denoted by (d; k)-digraphs. Miller and Fris showed that (2; k)-digraphs do not exist for k 3 . Subsequently, we gave a necessary condition ofthe existence of (3; k)-digraphs, namely, (3; k)-digraphs do not exist if k...
Graphs and Combinatorics 01/1998; 14:109-119. · 0.35 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Finite undirected graphs (possibly with loops, parallel edges) and finite groups are considered. We denote the radius and the diameter of a graph G by ρ(G) and δ(G), respectively. From the introduction: “The aim of this paper is to draw the attention to a construction well known in topological graph theory, the covering graph construction. Roughly speaking, it enables to ((blowup)) a given base graph to a larger graph.” The construction mentioned above is analyzed in detail in Part 2 (Voltage graphs and covering spaces) of the book [J. L. Gross and T. W. Tucker, Topological graph theory (1987; Zbl 0621.05013)]. We denote by G α the “large” graph obtained from a graph G and a group Γ by use of the assignment α:E(G)→Γ. The main result of the paper is the following one. Consider a group Γ and a generating system X of Γ such that X -1 =X and X contains no idempotent element. Let H be the Cayley graph Cay(Γ,X). Whenever G is a connected graph such that any degree exceeds |X| (in G), then there is an α such that δ(G α )<2(ρ(G)+δ(H)).
The Australasian Journal of Combinatorics [electronic only]. 01/1998; 18.