Results 1  10
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18
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 151 (1 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bo ..."
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Cited by 26 (4 self)
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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This survey aims to give an overview of the current stateoftheart of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
Planar lattice gases with nearestneighbour exclusion
"... We discuss the hardhexagon and hardsquare problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists. For this case we use the corner transfer matrix method to numerically evaluate the partition function per site ..."
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Cited by 18 (2 self)
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We discuss the hardhexagon and hardsquare problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists. For this case we use the corner transfer matrix method to numerically evaluate the partition function per site and density to 33 or more digits of accuracy. 1
Several constants arising in statistical mechanics Ann
 Comb
, 1999
"... Abstract. This is a brief survey of certain constants associated with random lattice models, including selfavoiding walks, polyominoes, the LenzIsing model, monomers and dimers, ice models, hard squares and hexagons, and percolation models. 1. ..."
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Cited by 4 (0 self)
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Abstract. This is a brief survey of certain constants associated with random lattice models, including selfavoiding walks, polyominoes, the LenzIsing model, monomers and dimers, ice models, hard squares and hexagons, and percolation models. 1.
Perfect factorisations of bipartite graphs and Latin squares without proper subrectangles
 J. Combin
, 1999
"... this paper. There are some important equivalence relations for Latin squares. Two squares are ..."
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Cited by 4 (1 self)
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this paper. There are some important equivalence relations for Latin squares. Two squares are
The search for pseudo orthogonal Latin squares of order six
 Des., Codes Crytogr
, 2000
"... 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. 1 Introduction Consider a Latin square S of order n.TheLatin square graph #ofS is defined on the entries of S, where two entries are ..."
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Cited by 3 (0 self)
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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. 1 Introduction Consider a Latin square S of order n.TheLatin square graph #ofS is defined on the entries of S, where two entries are adjacent whenever they are in the same row, in the same column, or carry the same symbol. It is wellknown and easily verified that if n # 3, # is strongly regular with parameters (n 2 , 3(n  1),n,6). If the Latin square S has an orthogonal mate S # ,then symbols of S # give a partition of the vertex set of #inton cocliques of size n, that is, a colouring of # with n colours. And vice versa, if # can be coloured with n colours, each colour class has to be a transversal of S, so the colour classes produce an orthogonal mate. A strongly regular graph with the same parameters as # is called a pseudo Latin square graph and if such a graph has chromatic number n, we speak of a pseudo or...
On the Dinitz conjecture and related conjectures
 Discrete Math
, 1995
"... We present previously unpublished elementary proofs by Dekker and Ottens (1991) and Boyce (private communication) of a special case of the Dinitz conjecture. We prove a special case of a related basis conjecture by Rota, and give a reformulation of Rota's conjecture using the Nullstellensatz. Finall ..."
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Cited by 3 (2 self)
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We present previously unpublished elementary proofs by Dekker and Ottens (1991) and Boyce (private communication) of a special case of the Dinitz conjecture. We prove a special case of a related basis conjecture by Rota, and give a reformulation of Rota's conjecture using the Nullstellensatz. Finally we give an asymptotic result on a related Latin square conjecture. 1.
Critical Sets in Latin Squares and Associated Structures
, 2001
"... A critical set in a Latin square of order n is a set of entries in an n x n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number ..."
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Cited by 2 (2 self)
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A critical set in a Latin square of order n is a set of entries in an n x n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of
Search and Enumeration Techniques for Incidence Structures
, 1998
"... This thesis investigates a number of probabilistic and exhaustive computational search techniques for the construction of a wide variety of combinatorial designs, and in particular, incidence structures. The emphasis is primarily from a computer science perspective, and focuses on the algorithmic de ..."
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Cited by 2 (0 self)
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This thesis investigates a number of probabilistic and exhaustive computational search techniques for the construction of a wide variety of combinatorial designs, and in particular, incidence structures. The emphasis is primarily from a computer science perspective, and focuses on the algorithmic development of the techniques, taking into account running time considerations and storage requirements. The search and enumeration techniques developed in this thesis have led to the discovery of a number of new results in the field of combinatorial design theory. Page ii Page iii Acknowledgments I would like to extend my sincere thanks to a number of people who have given me a great deal of assistance and support throughout the preparation of this thesis. Firstly, my supervisor Peter Gibbons. I am very grateful for the encouragement and guidance he has given to me. His remarkable enthusiasm and friendliness have helped to make this thesis a most enjoyable experience. My family, for their...
Some Simple 7Designs
 Combinatorial Designs and Related Structures, Proceedings of the First Pythagorean Conference, volume 245 of London Mathematical Society Lecture Notes
"... Some simple 7designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7(24; 8; 4): An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by ..."
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Cited by 1 (1 self)
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Some simple 7designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7(24; 8; 4): An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by known construction processes.