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The Moore bound for irregular graphs
 Graphs Combin
, 2001
"... What is the largest number of edges in a graph of order n and girth g? For dregular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an armative answer to an old open problem ([4] p.163, problem 10). ..."
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What is the largest number of edges in a graph of order n and girth g? For dregular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an armative answer to an old open problem ([4] p.163, problem 10).
Large Cayley Graphs and Digraphs with Small Degree and Diameter
, 1995
"... We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter. ..."
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We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.
Dynamic Cage Survey
, 2008
"... A (k, g)cage is a kregular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant const ..."
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A (k, g)cage is a kregular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.
A note on constructing large Cayley graphs of given degree and diameter by voltage assignments
, 1997
"... 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, which is to determine the largest order of a graph with given degree and diame ..."
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Cited by 6 (1 self)
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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, which is 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. the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertextransitive graphs of given degree and diameter....
On the existence of graphs of diameter two and defect two
 Discrete Mathematics 309 (2009
"... In the context of the degree/diameter problem, the ‘defect ’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n = d2 − 1. Only four extremal graphs of this type, referred to a ..."
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In the context of the degree/diameter problem, the ‘defect ’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n = d2 − 1. Only four extremal graphs of this type, referred to as (d, 2, 2)graphs, are known at present: two of degree d = 3 and one of degree d = 4 and 5, respectively. In this paper we prove, by using algebraic and spectral techniques, that for all values of the degree d within a certain range, (d, 2, 2)graphs do not exist. The enumeration of (d, 2, 2)graphs is equivalent to the search of binary symmetric matrices A fulfilling that AJn = dJn and A2 + A + (1 − d)In = Jn + B, where Jn denotes the allone matrix and B is the adjacency matrix of a union of graph cycles. In order to get the factorization of the characteristic polynomial of A in Q[x], we consider the polynomials Fi,d(x) = fi(x2 + x + 1 − d), where fi(x) denotes the minimal polynomial of the Gauss period ζi + ζi, being ζi a primitive ith root of unity. We formulate a conjecture on the irreducibility of Fi,d(x) in Q[x] and we show that its proof would imply the nonexistence of (d, 2, 2)graphs for any degree d> 5.
A Continuous Analogue of the Girth Problem
, 2001
"... Let A be the adjacency matrix of a dregular graph of order n and girth g, and d = 1 : : : n its eigenvalues. Then P n j=2 i j = nt i d i , for i = 0; 1; : : : ; g 1, where t i is the number of closed walks of length i on the dregular innite tree. Here we consider distributions on the rea ..."
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Let A be the adjacency matrix of a dregular graph of order n and girth g, and d = 1 : : : n its eigenvalues. Then P n j=2 i j = nt i d i , for i = 0; 1; : : : ; g 1, where t i is the number of closed walks of length i on the dregular innite tree. Here we consider distributions on the real line, whose ith moment is also nt i d i for all i = 0; 1; : : : ; g 1. We investigate distributional analogues of several extremal graph problems involving the parameters n, d, g and = max j 2 j; j n j. Surprisingly, perhaps, many similarities hold between the graphical and the distributional situations. Specically, we show in the case of distributions that the least possible n, given d; g is exactly the (trivial graphtheoretic) Moore bound. We also ask how small can be, given d; g and n, and improve the best known bound for graphs whose girth exceeds than their diameter. Institute of Mathematics, Hebrew University Jerusalem 91904 Israel alona@sunset.huji.ac.il y Institute of Computer Science, Hebrew University Jerusalem 91904 Israel shlomoh@cs.huji.ac.il z Institute of Computer Science, Hebrew University Jerusalem 91904 Israel nati@cs.huji.ac.il x Supported in part by grants from the USIsrael Binational Science Fund and from the Israeli Academy of Science. 1 1
Graphs of order two less than the Moore bound
 Discrete Mathematics 308 (2008
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The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
An Overview of the Degree/Diameter Problem
"... A wellknown fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diamete ..."
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A wellknown fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum outdegree d, respectively, maximum mixed degree) and diameter k. In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems. 1