Non-existence of bipartite graphs of diameter at least 4 and defect 2

- Pineda-Villavicencio, Guillermo

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Algebraic Combinatorics Vol. 34, no. 2 (2011), p. 163-182**Full Text:****Reviewed:****Description:**The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Î” and diameter D. Bipartite graphs of maximum degree Î”, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D=2,3,4 and 6, and for D=3,4,6, they have been constructed only for those values of Î” such that Î”-1 is a prime power. The scarcity of Moore bipartite graphs, together with the applications of such large topologies in the design of interconnection networks, prompted us to investigate what happens when the order of bipartite graphs misses the Moore bipartite bound by a small number of vertices. In this direction the first class of graphs to be studied is naturally the class of bipartite graphs of maximum degree Î”, diameter D, and two vertices less than the Moore bipartite bound (defect 2), that is, bipartite (Î”,D,-2)-graphs. For Î”â‰¥3 bipartite (Î”,2,-2)-graphs are the complete bipartite graphs with partite sets of orders Î” and Î”-2. In this paper we consider bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥3. Some necessary conditions for the existence of bipartite (Î”,3,-2)-graphs for Î”â‰¥3 are already known, as well as the non-existence of bipartite (Î”,D,-2)-graphs with Î”â‰¥3 and D=4,5,6,8. Furthermore, it had been conjectured that bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥4 do not exist. Here, using graph spectra techniques, we completely settle this conjecture by proving the non-existence of bipartite (Î”,D,-2)-graphs for all Î”â‰¥3 and all Dâ‰¥6. Â© 2010 Springer Science+Business Media, LLC.

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Journal of Algebraic Combinatorics Vol. 34, no. 2 (2011), p. 163-182**Full Text:****Reviewed:****Description:**The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree Î” and diameter D. Bipartite graphs of maximum degree Î”, diameter D and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if D=2,3,4 and 6, and for D=3,4,6, they have been constructed only for those values of Î” such that Î”-1 is a prime power. The scarcity of Moore bipartite graphs, together with the applications of such large topologies in the design of interconnection networks, prompted us to investigate what happens when the order of bipartite graphs misses the Moore bipartite bound by a small number of vertices. In this direction the first class of graphs to be studied is naturally the class of bipartite graphs of maximum degree Î”, diameter D, and two vertices less than the Moore bipartite bound (defect 2), that is, bipartite (Î”,D,-2)-graphs. For Î”â‰¥3 bipartite (Î”,2,-2)-graphs are the complete bipartite graphs with partite sets of orders Î” and Î”-2. In this paper we consider bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥3. Some necessary conditions for the existence of bipartite (Î”,3,-2)-graphs for Î”â‰¥3 are already known, as well as the non-existence of bipartite (Î”,D,-2)-graphs with Î”â‰¥3 and D=4,5,6,8. Furthermore, it had been conjectured that bipartite (Î”,D,-2)-graphs for Î”â‰¥3 and Dâ‰¥4 do not exist. Here, using graph spectra techniques, we completely settle this conjecture by proving the non-existence of bipartite (Î”,D,-2)-graphs for all Î”â‰¥3 and all Dâ‰¥6. Â© 2010 Springer Science+Business Media, LLC.

On graphs with cyclic defect or excess

- Delorme, Charles, Pineda-Villavicencio, Guillermo

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2010**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 17, no. 1 (2010), p.**Full Text:****Reviewed:****Description:**The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by Îµ are called graphs with defect or excess Îµ, respectively. While Moore graphs (graphs with Îµ = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation G_{d,k}(A) = J_{n}+B (G_{d,k}(A) = J_{n}- B), where A denotes the adjacency matrix of the graph in question, n its order, J_{n}the n Ã— n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and G_{d,k}(x) a polynomial with integer coefficients such that the matrix G_{d,k}(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O(64/3 d^{3/2}) for the number of graphs of odd degree d â‰¥ 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d â‰¥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the MÃ¶bius ladder on 8 vertices, no non-trivial graph of any degree â‰¥ 3 and cyclic defect or excess exists.

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2010**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 17, no. 1 (2010), p.**Full Text:****Reviewed:****Description:**The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by Îµ are called graphs with defect or excess Îµ, respectively. While Moore graphs (graphs with Îµ = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation G_{d,k}(A) = J_{n}+B (G_{d,k}(A) = J_{n}- B), where A denotes the adjacency matrix of the graph in question, n its order, J_{n}the n Ã— n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and G_{d,k}(x) a polynomial with integer coefficients such that the matrix G_{d,k}(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O(64/3 d^{3/2}) for the number of graphs of odd degree d â‰¥ 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d â‰¥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the MÃ¶bius ladder on 8 vertices, no non-trivial graph of any degree â‰¥ 3 and cyclic defect or excess exists.

On bipartite graphs of diameter 3 and defect 2

- Delorme, Charles, Jorgensen, Leif, Miller, Mirka, Pineda-Villavicencio, Guillermo

**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

**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

On graphs of defect at most 2

- Feria-Purón, Ramiro, Miller, Mirka, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 159, no. 13 (2011), p. 1331-1344**Full Text:****Reviewed:****Description:**In this paper we consider the degree/diameter problem, namely, given natural numbers Î”<2 and D<1, find the maximum number N(Î”,D) of vertices in a graph of maximum degree Î” and diameter D. In this context, the Moore bound M(Î”,D) represents an upper bound for N(Î”,D). Graphs of maximum degree Î”, diameter D and order M(Î”,D), called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Î”<2, diameter D<1 and order M(Î”,D)- with small >0, that is, (Î”,D,-)-graphs. The parameter is called the defect. Graphs of defect 1 exist only for Î”=2. When >1, (Î”,D,-)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in Feria-PurÃ³n and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Î”,D,-2)-graph with Î”<4 and D<4 is 2D. Second, and most important, we prove the non-existence of (Î”,D,-2)-graphs with even Î”<4 and D<4; this outcome, together with a proof on the non-existence of (4,3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-)-graphs with D<2 and 0â‰¤â‰¤2. Such a catalogue is only the second census of (Î”,D,-2)-graphs known at present, the first being that of (3,D,-)-graphs with D<2 and 0â‰¤â‰¤2 JÃ¸rgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of (Î”,D,-2)-graphs with odd Î”<5 and D<4, and the non-existence of (Î”,D,-2)-graphs with odd Î”<5 and D<5 such that Î”â‰¡0,2(modD). Finally, we conjecture that there are no (Î”,D,-2)-graphs with Î”<4 and D<4, and comment on some implications of our results for the upper bounds of N(Î”,D). Â© 2011 Elsevier B.V. All rights reserved.

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol. 159, no. 13 (2011), p. 1331-1344**Full Text:****Reviewed:****Description:**In this paper we consider the degree/diameter problem, namely, given natural numbers Î”<2 and D<1, find the maximum number N(Î”,D) of vertices in a graph of maximum degree Î” and diameter D. In this context, the Moore bound M(Î”,D) represents an upper bound for N(Î”,D). Graphs of maximum degree Î”, diameter D and order M(Î”,D), called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Î”<2, diameter D<1 and order M(Î”,D)- with small >0, that is, (Î”,D,-)-graphs. The parameter is called the defect. Graphs of defect 1 exist only for Î”=2. When >1, (Î”,D,-)-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in Feria-PurÃ³n and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a (Î”,D,-2)-graph with Î”<4 and D<4 is 2D. Second, and most important, we prove the non-existence of (Î”,D,-2)-graphs with even Î”<4 and D<4; this outcome, together with a proof on the non-existence of (4,3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-)-graphs with D<2 and 0â‰¤â‰¤2. Such a catalogue is only the second census of (Î”,D,-2)-graphs known at present, the first being that of (3,D,-)-graphs with D<2 and 0â‰¤â‰¤2 JÃ¸rgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of (Î”,D,-2)-graphs with odd Î”<5 and D<4, and the non-existence of (Î”,D,-2)-graphs with odd Î”<5 and D<5 such that Î”â‰¡0,2(modD). Finally, we conjecture that there are no (Î”,D,-2)-graphs with Î”<4 and D<4, and comment on some implications of our results for the upper bounds of N(Î”,D). Â© 2011 Elsevier B.V. All rights reserved.

On bipartite graphs of defect at most 4

- Feria-Purón, Ramiro, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol.160, no.1-2 (2011), p.140-154**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Î” â‰¥ 2 and D â‰¥ 2, find the maximum number Nb (Î”, D) of vertices in a bipartite graph of maximum degree Î” and diameter D. In this context, the Moore bipartite bound Mb (Î”, D) represents an upper bound for Nb (Î”, D). Bipartite graphs of maximum degree Î”, diameter D and order Mb (Î”, D)-called Moore bipartite graphs-have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Î” â‰¥ 2, diameter D â‰¥ 2 and order Mb (Î”, D) - Îµ{lunate} with small Îµ{lunate} > 0, that is, bipartite (Î”, D, - Îµ{lunate})-graphs. The parameter Îµ{lunate} is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if Î” â‰¥ 3 and D â‰¥ 3, they may only exist for D = 3. However, when Îµ{lunate} > 2 bipartite (Î”, D, - Îµ{lunate})-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (Î”, D, - 4)-graphs; the complete catalogue of bipartite (3, D, - Îµ{lunate})-graphs with D â‰¥ 2 and 0 â‰¤ Îµ{lunate} â‰¤ 4; the complete catalogue of bipartite (Î”, D, - Îµ{lunate})-graphs with Î” â‰¥ 2, 5 â‰¤ D â‰¤ 187 (D â‰ 6) and 0 â‰¤ Îµ{lunate} â‰¤ 4; a proof of the non-existence of all bipartite (Î”, D, - 4)-graphs with Î” â‰¥ 3 and odd D â‰¥ 5. Finally, we conjecture that there are no bipartite graphs of defect 4 for Î” â‰¥ 3 and D â‰¥ 5, and comment on some implications of our results for the upper bounds of Nb (Î”, D). Â© 2011 Elsevier B.V. All rights reserved.

**Authors:**Feria-Purón, Ramiro , Pineda-Villavicencio, Guillermo**Date:**2011**Type:**Text , Journal article**Relation:**Discrete Applied Mathematics Vol.160, no.1-2 (2011), p.140-154**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Î” â‰¥ 2 and D â‰¥ 2, find the maximum number Nb (Î”, D) of vertices in a bipartite graph of maximum degree Î” and diameter D. In this context, the Moore bipartite bound Mb (Î”, D) represents an upper bound for Nb (Î”, D). Bipartite graphs of maximum degree Î”, diameter D and order Mb (Î”, D)-called Moore bipartite graphs-have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Î” â‰¥ 2, diameter D â‰¥ 2 and order Mb (Î”, D) - Îµ{lunate} with small Îµ{lunate} > 0, that is, bipartite (Î”, D, - Îµ{lunate})-graphs. The parameter Îµ{lunate} is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if Î” â‰¥ 3 and D â‰¥ 3, they may only exist for D = 3. However, when Îµ{lunate} > 2 bipartite (Î”, D, - Îµ{lunate})-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (Î”, D, - 4)-graphs; the complete catalogue of bipartite (3, D, - Îµ{lunate})-graphs with D â‰¥ 2 and 0 â‰¤ Îµ{lunate} â‰¤ 4; the complete catalogue of bipartite (Î”, D, - Îµ{lunate})-graphs with Î” â‰¥ 2, 5 â‰¤ D â‰¤ 187 (D â‰ 6) and 0 â‰¤ Îµ{lunate} â‰¤ 4; a proof of the non-existence of all bipartite (Î”, D, - 4)-graphs with Î” â‰¥ 3 and odd D â‰¥ 5. Finally, we conjecture that there are no bipartite graphs of defect 4 for Î” â‰¥ 3 and D â‰¥ 5, and comment on some implications of our results for the upper bounds of Nb (Î”, D). Â© 2011 Elsevier B.V. All rights reserved.

On large bipartite graphs of diameter 3

- Feria-Purón, Ramiro, Miller, Mirka, Pineda-Villavicencio, Guillermo

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers dâ‰¥2 and Dâ‰¥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-PurÃ³n and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.**Description:**2003011037

**Authors:**Feria-Purón, Ramiro , Miller, Mirka , Pineda-Villavicencio, Guillermo**Date:**2013**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 313, no. 4 (2013), p. 381-390**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**We consider the bipartite version of the degree/diameter problem, namely, given natural numbers dâ‰¥2 and Dâ‰¥2, find the maximum number N b(d,D) of vertices in a bipartite graph of maximum degree d and diameter D. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D) pairs. This paper is a follow-up of our earlier paper (Feria-PurÃ³n and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,-4)-graphs (that is, bipartite graphs of order M b(d,D)-4) was carried out. Here we first present some structural properties of bipartite (d,3,-4)-graphs, and later prove that there are no bipartite (7,3,-4)-graphs. This result implies that the known bipartite (7,3,-6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner-Loz graph after its first discoverers Paul Hafner and Eyal Loz. The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,-4)-graph, and the non-existence of bipartite (6,3,-4)-graphs. In addition, we discover at least one new largest known bipartite-and also vertex-transitive-graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3). © 2012 Elsevier B.V. All rights reserved.**Description:**2003011037

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