### Super edge-antimagicness for a class of disconnected graphs

**Authors:**Dafik , Miller, Mirka , Ryan, Joe , Baca, Martin**Date:**2006**Type:**Text , Conference paper**Relation:**Paper presented at AWOCA 2006, 17th Australasian Workshop on Combinatorial Algorithms, Uluru, Australia : 13th July, 2006 p. 67-75**Full Text:**false**Reviewed:****Description:**E1**Description:**2003001916

### On the nonexistence of graphs of diameter 2 and defect 2

**Authors:**Miller, Mirka , Nguyen, Minh Hoang , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**The Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 71, no. (2009), p. 5-20**Full Text:**false**Reviewed:****Description:**In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree d and d² + 1 vertices), and found that such graphs exist only for d = 2; 3; 7 and possibly 57. In 1980, Erdös et al., using eigenvalue analysis, showed that, with the exception of C4, there are no graphs of diameter 2, maximum degree d and d² vertices. In this paper, we show that graphs of diameter 2, maximum degree d and d² - 1 vertices do not exist for most values of d with d ≥ 6, and conjecture that they do not exist for any d ≥ 6.**Description:**2003007893

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

**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

### Graphs of order two less than the Moore bound

**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

### Delta-optimum exclusive sum labeling of certain graphs with radius one

**Authors:**Tuga, Mauritsius , Miller, Mirka**Date:**2005**Type:**Text , Journal article**Relation:**Lecture Notes in Computer Science Vol. 3330, no. (2005), p. 216-225**Full Text:**false**Reviewed:****Description:**A mapping**Description:**C1**Description:**2003001413

### New largest known graphs of diameter 6

**Authors:**Pineda-Villavicencio, Guillermo , Gómez, José , Miller, Mirka , Pérez-Rosés, Hebert**Date:**2009**Type:**Text , Journal article**Relation:**Networks Vol. 53, no. 4 (2009), p. 315-328**Full Text:****Reviewed:****Description:**In the pursuit of obtaining largest graphs of given maximum degree**Description:**2003007890

### On graphs of maximum size with given girth and order

**Authors:**Miller, Mirka , Lin, Yuqing , Brankovic, Ljiljana , Tang, Jianmin**Date:**2006**Type:**Text , Conference paper**Relation:**Paper presented at AWOCA 2006, 17th Australasian Workshop on Combinatorial Algorithms, Uluru, Australia : 13th July, 2006**Full Text:**false**Reviewed:****Description:**E1**Description:**2003001918

### On network security and internet vulnerability

**Authors:**Miller, Mirka , Patel, Deval , Patel, Keyurkumar**Date:**2005**Type:**Text , Conference paper**Relation:**Paper presented at IADIS International Conference on Applied Computing 2005, Volume II, Algarve, Portugal : 22nd - 25th February, 2005**Full Text:**false**Reviewed:****Description:**E1**Description:**2003001409

### Antimagic labeling of disjoint union of s-crowns

**Authors:**Baca, Martin , Dafik , Miller, Mirka , Ryan, Joe**Date:**2009**Type:**Text , Journal article**Relation:**Utilitas Mathematica Vol. 79, no. (2009), p. 193-205**Full Text:**false**Reviewed:****Description:**A graph G is called (a, d)-edge-antimagic total if it admits a labeling of the vertices and edges by pairwise distinct integers of 1,2,..., |V(G)| + |E(G)| such that the edge-weights, w(uÏ…) = f(u) + f(Ï…) + f(uÏ…), uv âˆˆ E(G), form an arithmetic sequence with the first term a and common difference d. Such a graph G is called super if the smallest possible labels appear on the vertices. A construction of super (a, d)-edge-antimagic total labelings of some disconnected graphs are described.

### On d-antimagic labelings of prisms

**Authors:**Lin, Yuqing , Slamin, , Baca, Martin , Miller, Mirka**Date:**2004**Type:**Text , Journal article**Relation:**Ars Combinatoria: A Canadian Journal of Combinatorics Vol. 72, no. (2004), p. 65-76**Full Text:**false**Reviewed:****Description:**C1**Description:**2003000907

### On the degrees of a strongly vertex-magic graph

**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

### Super vertex-magic total labelings of graphs

**Authors:**MacDougall, James , Miller, Mirka , Sugeng, Kiki Ariyanti**Date:**2004**Type:**Text , Conference paper**Relation:**Paper presented at AWOCA 2004: Fifteenth Australasian Workshop on Combinatorial Algorithms, Ballina, New South Wales : 6th - 9th July, 2004 p. 222–229**Full Text:**false**Reviewed:****Description:**E1**Description:**2003000902

### On non-polynomiality of XOR over Zn2

**Authors:**Grosek, Otokar , Miller, Mirka , Ryan, Joe**Date:**2004**Type:**Text , Journal article**Relation:**Tatra Mountains Mathematical Publications Vol. 29, no. (2004), p. 183-191**Full Text:**false**Reviewed:****Description:**C1**Description:**2003000905

### Two new families of large compound graphs

**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

### Characterization of eccentric digraphs

**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

### Cordial labelling of butterfly networks and mesh of trees

**Authors:**Miller, Mirka , Rajan, Bharati , Rajasingh, Indra , Manuel, Paul**Date:**2006**Type:**Text , Conference paper**Relation:**Paper presented at AWOCA 2006, 17th Australasian Workshop on Combinatorial Algorithms, Uluru, Australia : 13th July, 2006**Full Text:**false**Reviewed:****Description:**E1**Description:**2003001646

### On bipartite graphs of diameter 3 and defect 2

**Authors:**Delorme, Charles , Jorgensen, Leif , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**Journal of Graph Theory Vol. 61, no. 4 (2009), p. 271-288**Full Text:****Reviewed:****Description:**We consider bipartite graphs of degree A<2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (â–³,3, -2) -graphs. We prove the uniqueness of the known bipartite (3, 3, -2) -graph and bipartite (4, 3, -2)-graph. We also prove several necessary conditions for the existence of bipartite (â–³,3, -2) - graphs. The most general of these conditions is that either â–³ or â–³-2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when â–³ = 6 and â–³ = 9, we prove the non-existence of the corresponding bipartite (â–³,3,-2)-graphs, thus establishing that there are no bipartite (â–³,3, -2)-graphs, for 5

### Further results in d-antimagic labelings of antiprisms

**Authors:**Lin, Yuqing , Ahmad, Abeed , Miller, Mirka , Sugeng, Kiki Ariyanti , Baca, Martin**Date:**2004**Type:**Text , Conference paper**Relation:**Paper presented at AWOCA 2004: Fifteenth Australasian Workshop on Combinatorial Algorithms, Ballina, New South Wales : 6-9th July, 2004**Full Text:**false**Reviewed:****Description:**E1**Description:**2003000900

### Antimagic labelings of grids

**Authors:**Baca, Martin , Lin, Yuqing , Miller, Mirka**Date:**2007**Type:**Text , Journal article**Relation:**Utilitas Mathematica Vol. 72, no. (2007), p. 65-75**Full Text:**false**Reviewed:****Description:**In this paper we deal with the problem of labeling the vertices, edges and faces of a grid graph by the consecutive integers from 1 to |V| + |E| + |F| in such a way that the label of a face and the labels of the vertices and edges surrounding that face all together add up to a weight of that face. These face weights then form an arithmetic progression with common difference d.**Description:**C1**Description:**2003004808

### On irregular total labellings

**Authors:**Baca, Martin , Jendrol, Stanislav , Miller, Mirka , Ryan, Joe**Date:**2007**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 307, no. 11-12 (May 2007), p. 1378-1388**Full Text:****Reviewed:****Description:**Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved. (c) 2006 Elsevier B.V. All rights reserved.**Description:**C1**Description:**2003004909