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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
A Note on Large Graphs of Diameter Two and Given Maximum Degree
"... Let vt(d; 2) be the largest order of a vertextransitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d ..."
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Cited by 19 (5 self)
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Let vt(d; 2) be the largest order of a vertextransitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertextransitive nonCayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the HoffmanSingleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The wellknown degree/diameter problem asks for determining the largest possible number n(d; k) of vertic...
The Chromatic Number of Graph Powers
, 2000
"... Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? ..."
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Cited by 15 (0 self)
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Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? Our (somewhat surprising) answer is that for g = 3; 4; 5 or 6 this maximum is (1 + o(1))d 2 (where the o(1) term tends to 0 as d tends to infinity), whereas for all g 7, this maximum is of order d 2 = log d. To state this result more precisely, define, for every two integers d 2 and g<F9.9
Bounds on Graph Eigenvalues
"... We refute, improve or amplify some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n ≥ 2, maximum degree ∆, and girth at least 5, then the maximum eigenvalue µ (G) of the adjacency matrix of G satisfies µ (G) ≤ min { ∆, √ n − 1}. and Also if G is a graph ..."
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Cited by 12 (8 self)
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We refute, improve or amplify some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n ≥ 2, maximum degree ∆, and girth at least 5, then the maximum eigenvalue µ (G) of the adjacency matrix of G satisfies µ (G) ≤ min { ∆, √ n − 1}. and Also if G is a graph of order n ≥ 2 with dominating number γ (G) = γ, then n if γ = 1 λ2 (G) ≤ n − γ if γ ≥ 2, λn (G) ≥ ⌈n/γ ⌉,
Digraphs of degree 3 and order close to the Moore bound
, 1995
"... It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are no digraphs of order `close' to, i.e., one less than, Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 ..."
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Cited by 9 (6 self)
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It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are no digraphs of order `close' to, i.e., one less than, Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore 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 many values of the diameter. Keywords  digraphs, Moore bound, diameter, degree. 1. Introduction By a digraph we mean a structure G = (V; A) where V (G) is a nonempty set of distinct elements called vertices; and A(G) is a set of ordered pairs (u; v) of distinct vertices u; v 2 V called arcs. The order of a digraph G is the number of vertices in G, i.e., jV (G)j. An inneighbour of a vertex v in a digraph G is a vertex u such that (u; v) 2 G. Similarly, an outneighbour of a vertex v is a v...
Regular Digraphs of Diameter 2 and Maximum Order
, 1994
"... It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [16] or [4]). For degree 2, it has been shown that for diameter k 3 there are no digraphs of order `close' to, i.e., one less than, the Moore bound [14]. For diameter 2, it is known that digraphs close to Moore bou ..."
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Cited by 8 (6 self)
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It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [16] or [4]). For degree 2, it has been shown that for diameter k 3 there are no digraphs of order `close' to, i.e., one less than, the Moore bound [14]. For diameter 2, it is known that digraphs close to Moore bound exist for any degree because the line digraphs of complete digraphs are an example of such digraphs. However, it is not known whether these are the only digraphs close to Moore digraphs. In this paper, we shall consider the general case of directed graphs of diameter 2, degree d 3 and with the number of vertices n = d + d 2 , that is, one less than the Moore bound. Using the eigenvalues of the corresponding adjacency matrices we give a number of necessary conditions for the existence of such digraphs. Furthermore, for degree 3 we prove that there are no digraphs close to Moore bound other than the line digraph of K 4 . Keywords  digraphs, Moore bound, diameter, degree. 1. Introduc...
On the structure of digraphs with order close to the Moore bound
, 1996
"... The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k . It is known that digraphs of order M d;k do not exist for d ? 1 and k ? 1 ([24] or [6]). In this paper we study digraphs of degree d, diameter k and order M d;k \Gamma 1, denoted by (d; k)digraphs ..."
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Cited by 7 (5 self)
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The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k . It is known that digraphs of order M d;k do not exist for d ? 1 and k ? 1 ([24] or [6]). In this paper we study digraphs of degree d, diameter k and 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 [22]. Subsequently, we gave a necessary condition of the existence of (3; k)digraphs, namely, (3; k)digraphs do not exist if k is odd or if k + 1 does not divide 9 2 (3 k \Gamma 1) [3]. The (d; 2)digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d; k)digraphs. In particular, for d; k 3, we show that a (d; k)digraph contains either no cycle of length k or exactly one cycle of length k. 1 Introduction By a digraph we mean a structure G = (V; A) where V (G) is a nonempty set of elements called vertices; and A(G) is a set of ordered pairs (u; v) of disti...
Maximum degree in graphs of diameter 2
 Networks
, 1980
"... It is well known that there are at most four Moore graphs of diameter 2, i.e., graphs of diameter 2, maximum degree d, and d 2 + 1 vertices. The purpose of this paper is to prove that with the exception of C4, there are no graphs of diameter 2, of maximum degree d, and with d 2 vertices. ..."
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Cited by 6 (0 self)
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It is well known that there are at most four Moore graphs of diameter 2, i.e., graphs of diameter 2, maximum degree d, and d 2 + 1 vertices. The purpose of this paper is to prove that with the exception of C4, there are no graphs of diameter 2, of maximum degree d, and with d 2 vertices.
Domination in planar graphs with small diameter
 2002) 1–25. the electronic journal of combinatorics 10 (2003), #N9 5
"... MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large do ..."
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Cited by 5 (1 self)
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MacGillivray and Seyffarth (J. Graph Theory 22 (1996), 213–229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a unique planar graph of diameter two with domination number three, and all other planar graphs of diameter two have domination number at most two. We also prove that every planar graph of diameter three and of radius two has domination number at most six. We then show that every sufficiently large planar graph of diameter three has domination number at most seven. Analogous results for other surfaces are discussed. 1