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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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Cited by 48 (7 self)
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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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Cited by 10 (0 self)
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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.
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 4 (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....
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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Cited by 2 (1 self)
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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
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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Cited by 1 (1 self)
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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...