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7Lin, Yuqing
5Balbuena, Camino
3Marcote, Xavier
3Sugeng, Kiki Ariyanti
2Barker, Ewan
2Baskoro, Edy
2Gimbert, Joan
2Ryan, Joe
1Cholily, Yus Mochamad
1Das, K. C.
1Jiang, T.
1Lopez, Nacho
1Marti, J. Gomez
1Nguyen, Minh Hoang
1Rodger, Chris
1Simanjuntak, Rinovia
1Siran, Jozef
1Slamin,
1Sutton, Martin

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110101 Pure Mathematics
5Numerical methods
4Connectivity
4Moore bound
4Number theory
4Set theory
4Theorem proving
3Integer programming
3Problem solving
2(k,g)-cage
20802 Computation Theory and Mathematics
2Digraphs
1(Δ, D)-problem
10103 Numerical and Computational Mathematics
1Algorithms
1Almost Moore digraph
1Bijection
1Boundary value problems
1Bounded diameter

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Enumerations of vertex orders of almost Moore digraphs with selfrepeats

- Baskoro, Edy, Cholily, Yus Mochamad, Miller, Mirka

**Authors:**Baskoro, Edy , Cholily, Yus Mochamad , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 1 (2008), p. 123-128**Full Text:**false**Reviewed:****Description:**An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r (u) = v, such that there are two walks of length ≤ k from u to v. The smallest positive integer p such that the composition rp (u) = u is called the order of u. If the order of u is 1 then u is called a selfrepeat. It is known that if G is an almost Moore digraph of diameter k ≥ 3 then G contains exactly k selfrepeats or none. In this paper, we propose an exact formula for the number of all vertex orders in an almost Moore digraph G containing selfrepeats, based on the vertex orders of the out-neighbours of any selfrepeat vertex. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

Graphs of order two less than the Moore bound

- Miller, Mirka, Simanjuntak, Rinovia

**Authors:**Miller, Mirka , Simanjuntak, Rinovia**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 13 (2008), p. 2810-2821**Full Text:**false**Reviewed:****Description:**The Moore bound for a directed graph of maximum out-degree d and diameter k is Md,k=1+d+d2++dk. It is known that digraphs of order Md,k (Moore digraphs) do not exist for d>1 and k>1. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is . Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter k=2 are unique and that mixed Moore graphs of diameter k3 do not exist.**Description:**C1

HSAGA and its application for the construction of near-Moore digraphs

- Tang, Jianmin, Miller, Mirka, Lin, Yuqing

**Authors:**Tang, Jianmin , Miller, Mirka , Lin, Yuqing**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Discrete Algorithms Vol. 6, no. 1 (2008), p. 73-84**Full Text:**false**Reviewed:****Description:**The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. This paper deals with directed graphs. General upper bounds, called Moore bounds, exist for the largest possible order of such digraphs of maximum degree d and given diameter k. It is known that simulated annealing and genetic algorithm are effective techniques to identify global optimal solutions. This paper describes our attempt to build a Hybrid Simulated Annealing and Genetic Algorithm (HSAGA) that can be used to construct large digraphs. We present our new results obtained by HSAGA, as well as several related open problems. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

On consecutive edge magic total labeling of graphs

- Sugeng, Kiki Ariyanti, Miller, Mirka

**Authors:**Sugeng, Kiki Ariyanti , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Journal of Discrete Algorithms Vol. 6, no. 1 (2008), p. 59-65**Full Text:**false**Reviewed:****Description:**Let G = (V, E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection**Description:**C1

On the connectivity of (k, g)-cages of even girth

- Lin, Yuqing, Balbuena, Camino, Marcote, Xavier, Miller, Mirka

**Authors:**Lin, Yuqing , Balbuena, Camino , Marcote, Xavier , Miller, Mirka**Date:**2008**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 308, no. 15 (2008), p. 3249-3256**Full Text:**false**Reviewed:****Description:**A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even. © 2007 Elsevier B.V. All rights reserved.**Description:**C1

A lower bound on the order of regular graphs with given girth pair

- Balbuena, Camino, Jiang, T., Lin, Yuqing, Marcote, Xavier, Miller, Mirka

**Authors:**Balbuena, Camino , Jiang, T. , Lin, Yuqing , Marcote, Xavier , Miller, Mirka**Date:**2007**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 55, no. 2 (2007), p. 153-163**Full Text:**false**Reviewed:****Description:**The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 (1983), 209-218]. A (**Description:**C1**Description:**2003004727

- Nguyen, Minh Hoang, Miller, Mirka, Gimbert, Joan

**Authors:**Nguyen, Minh Hoang , Miller, Mirka , Gimbert, Joan**Date:**2007**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 307, no. 7-8 (2007), p. 964-970**Full Text:**false**Reviewed:****Description:**The Moore bound for a directed graph of maximum out-degree d and diameter k is M**Description:**C1**Description:**2003005024

All (k;g)-cages are edge-superconnected

- Lin, Yuqing, Miller, Mirka, Balbuena, Camino, Marcote, Xavier

**Authors:**Lin, Yuqing , Miller, Mirka , Balbuena, Camino , Marcote, Xavier**Date:**2006**Type:**Text , Journal article**Relation:**Networks Vol. 47, no. 2 (2006), p. 102-110**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is k-regular graph with girth g and with the least possible number of vertices. In this article we prove that (k;g)-cages are edge-superconnected if g is even. Earlier, Marcote and Balbuena proved that (k;g)-cages are edge-superconnected if g is odd [Networks 43 (2004), 54-59]. Combining our results, we conclude that all (k;g)-cages are edge-superconnected. © 2005 Wiley Periodicals, Inc.**Description:**C1**Description:**2003001830

Characterization of eccentric digraphs

- Gimbert, Joan, Lopez, Nacho, Miller, Mirka, Ryan, Joe

**Authors:**Gimbert, Joan , Lopez, Nacho , Miller, Mirka , Ryan, Joe**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 2 (2006), p. 210-219**Full Text:**false**Reviewed:****Description:**The eccentric digraph ED(G) of a digraph G represents the binary relation, defined on the vertex set of G, of being 'eccentric'; that is, there is an arc from u to v in ED(G) if and only if v is at maximum distance from u in G. A digraph G is said to be eccentric if there exists a digraph H such that G=ED(H). This paper is devoted to the study of the following two questions: what digraphs are eccentric and when the relation of being eccentric is symmetric. We present a characterization of eccentric digraphs, which in the undirected case says that a graph G is eccentric iff its complement graph G is either self-centered of radius two or it is the union of complete graphs. As a consequence, we obtain that all trees except those with diameter 3 are eccentric digraphs. We also determine when ED(G) is symmetric in the cases when G is a graph or a digraph that is not strongly connected. Crown Copyright © 2006 Published by Elsevier B.V. All rights reserved.**Description:**C1**Description:**2003001601

- Balbuena, Camino, Barker, Ewan, Lin, Yuqing, Miller, Mirka, Sugeng, Kiki Ariyanti

**Authors:**Balbuena, Camino , Barker, Ewan , Lin, Yuqing , Miller, Mirka , Sugeng, Kiki Ariyanti**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 16 (2006), p. 1817-1829**Full Text:**false**Reviewed:****Description:**Let G be a graph of order n and size e. A vertex-magic total labeling is an assignment of the integers 1, 2, ..., n + e to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic number of G. Such a labeling is a-vertex consecutive magic if the set of the labels of the vertices is { a + 1, a + 2, ..., a + n }, and is b-edge consecutive magic if the set of labels of the edges is { b + 1, b + 2, ..., b + e }. In this paper we prove that if an a-vertex consecutive magic graph has isolated vertices then the order and the size satisfy (n - 1)**Description:**C1**Description:**2003001604

On the degrees of a strongly vertex-magic graph

- Balbuena, Camino, Barker, Ewan, Das, K. C., Lin, Yuqing, Miller, Mirka, Ryan, Joe, Slamin,, Sugeng, Kiki Ariyanti, Tkac, M.

**Authors:**Balbuena, Camino , Barker, Ewan , Das, K. C. , Lin, Yuqing , Miller, Mirka , Ryan, Joe , Slamin, , Sugeng, Kiki Ariyanti , Tkac, M.**Date:**2006**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 306, no. 6 (2006), p. 539-551**Full Text:**false**Reviewed:****Description:**Let G=(V,E) be a finite graph, where |V|=n≥2 and |E|=e≥1. A vertex-magic total labeling is a bijection λ from V∪E to the set of consecutive integers {1,2,...,n+e} with the property that for every v∈V, λ(v)+∑w∈N(v)λ(vw)=h for some constant h. Such a labeling is strong if λ(V)={1,2,...,n}. In this paper, we prove first that the minimum degree of a strongly vertex-magic graph is at least two. Next, we show that if 2e≥10n2-6n+1, then the minimum degree of a strongly vertex-magic graph is at least three. Further, we obtain upper and lower bounds of any vertex degree in terms of n and e. As a consequence we show that a strongly vertex-magic graph is maximally edge-connected and hamiltonian if the number of edges is large enough. Finally, we prove that semi-regular bipartite graphs are not strongly vertex-magic graphs, and we provide strongly vertex-magic total labeling of certain families of circulant graphs. © 2006 Elsevier B.V. All rights reserved**Description:**C1**Description:**2003001603

Two new families of large compound graphs

- Marti, J. Gomez, Miller, Mirka

**Authors:**Marti, J. Gomez , Miller, Mirka**Date:**2006**Type:**Text , Journal article**Relation:**Networks Vol. 47, no. 3 (2006), p. 140-146**Full Text:**false**Reviewed:****Description:**A question of special interest in graph theory is the design of large graphs. Specifically, we want to find constructions of graphs with order as large as possible for a given degree A and diameter D. Two generalizations of two large compound graphs are proposed in this article. Three particular cases of these families of graphs presented here allow us to improve the order for the entries (15, 7), (13, 10), and (15, 10) in the table of the largest known (Δ, D)-graphs. © 2006 Wiley Periodicals, Inc.**Description:**C1**Description:**2003001599

All (k;g)-cages are k-edge-connected

- Lin, Yuqing, Miller, Mirka, Rodger, Chris

**Authors:**Lin, Yuqing , Miller, Mirka , Rodger, Chris**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 48, no. 3 (2005), p. 219-227**Full Text:**false**Reviewed:****Description:**A (k;g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)-cages are k-edge-connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)-cages are k-edge-connected if g is odd. Combining our results, we conclude that the (k;g)-cages are k-edge-connected. © 2005 wiley Periodicals, Inc.**Description:**C1

Complete characterization of almost moore digraphs of degree three

- Baskoro, Edy, Miller, Mirka, Siran, Jozef, Sutton, Martin

**Authors:**Baskoro, Edy , Miller, Mirka , Siran, Jozef , Sutton, Martin**Date:**2005**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 48, no. 2 (2005), p. 112-126**Full Text:**false**Reviewed:****Description:**It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc.**Description:**C1**Description:**2003000904

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