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Complete catalogue of graphs of maximum degree 3 and defect at most 4

- Miller, Mirka, Pineda-Villavicencio, Guillermo

**Authors:**Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 157, no. 13 (2009), p. 2983-2996**Full Text:****Reviewed:****Description:**We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.

**Authors:**Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2009**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 157, no. 13 (2009), p. 2983-2996**Full Text:****Reviewed:****Description:**We consider graphs of maximum degree 3, diameter D≥2 and at most 4 vertices less than the Moore bound M3,D, that is, (3,D,−)-graphs for ≤4. We prove the non-existence of (3,D,−4)-graphs for D≥5, completing in this way the catalogue of (3,D,−)-graphs with D≥2 and ≤4. Our results also give an improvement to the upper bound on the largest possible number N3,D of vertices in a graph of maximum degree 3 and diameter D, so that N3,D≤M3,D−6 for D≥5. Copyright Elsevier.

On the nonexistence of graphs of diameter 2 and defect 2

- Miller, Mirka, Nguyen, Minh Hoang, Pineda-Villavicencio, Guillermo

**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

Moore graphs and beyond : A survey of the degree/diameter problem

**Authors:**Miller, Mirka , Siran, Jozef**Date:**2005**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. DS14, no. (2005), p. 1-61**Full Text:**false**Reviewed:****Description:**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 state-of-the-art 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 Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.**Description:**C1**Description:**2003001407

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