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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 191 (2 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 45 (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 19 (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
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, Designs Codes Crypt
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Several constants arising in statistical mechanics
, 1999
"... 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. ..."
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Cited by 4 (0 self)
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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.
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 3 (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...
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. F ..."
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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.
On the classification of Hadamard matrices of order 32
, 2009
"... All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of inequivalent Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that ..."
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Cited by 2 (0 self)
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All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of inequivalent Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard matrices of one type and 26,369 such matrices of another type. Based on experience with the classification of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insignificant.