On some classes of Deza graphs Deza graphs without 3-cocliques - PowerPoint PPT Presentation

On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs On some classes of Deza graphs Deza graphs without 3-cocliques Line graphs V.V. Kabanov 1 Deza graphs obtained from vvk@imm.uran.ru srg L.V. Shalaginov 1 , 2 Lists of Deza graps and 44sh@mail.ru Cayley-Deza graphs Deza graphs with 1 Institute of mathematics and mechanics UB RAS disconnected S.Kovalevskoy, 16, Ekaterinburg, 620219, Russia second 2 Chelyabinsk state university neighborhood Br. Kashirinykh, 129, Chelyabinsk, 454001, Russia 2015

Contents On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs Deza graphs Deza graphs without 3-cocliques Line graphs Deza graphs without 3-cocliques Deza graphs obtained from srg Line graphs Lists of Deza graps and Cayley-Deza graphs Deza graphs Deza graphs obtained from srg with disconnected second neighborhood Lists of Deza graps and Cayley-Deza graphs Deza graphs with disconnected second neighborhood

Definition On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs We consider the following generalization of strongly regular graphs. Deza graphs without 3-cocliques Definition Line graphs Let v, k, b and a be integers such that 0 ≤ a ≤ b ≤ k < v. A graph Deza graphs Γ is a Deza graph with parameters ( v, k, b, a ) if obtained from srg ◮ Γ has exactly v vertices; Lists of Deza graps and ◮ for any vertex u in Γ its neighbourhood Γ( u ) has exactly k Cayley-Deza graphs vertices; Deza graphs ◮ for any two different vertices u, w in Γ the intersection with disconnected Γ( u ) ∩ Γ( w ) takes on one of two values b and a . second neighborhood The only difference between a strongly regular graph and a Deza graph is that the size of Γ( u ) ∩ Γ( w ) does not depend on adjacency u and w .

Some history On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs Deza graphs without These graphs were introduced by Antoine and Michel Deza. 3-cocliques Line graphs A. Deza and M. Deza Deza graphs The ridge graph of the metric polytope and some relatives obtained from srg Polytopes: Abstract, convex and computational Lists of Deza T. Bisztriczky et al. (Editors). NATO ASI Series, Kluwer Academic. graps and Cayley-Deza 1994, P. 359-372. graphs Deza graphs In the case a = 0 a Deza graph can have the diameter greater than with disconnected 2 , then this case Deza graph is considered separately. A strictly Deza second neighborhood graph SDG is a Deza graph which is not strongly regular SRG and has diameter 2 .

Adjacency matrices On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs Deza graphs without Let M be the adjacency matrix a graph Γ . Then Γ is a Deza graph 3-cocliques with parameters ( v, k, b, a ) if and only if Line graphs Deza graphs M 2 = aA + bB + kI obtained from srg Lists of Deza for some (0 , 1) -matrices A and B such that A + B + I = J , the all graps and Cayley-Deza ones matrix. Note that Γ is a strongly regular graph if and only if A graphs or B is M . As usual we used parameters ( v, k, λ, µ ) for a strongly Deza graphs with regular graph. So we have the matrix equation disconnected second M 2 = λM + µ ( J − M − I ) + kI. neighborhood

Some history On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs Deza graphs without 3-cocliques The study of strongly regular graphs has a long history, and the Line graphs study of strictly Deza graphs started relatively recently. Significant Deza graphs obtained from results for strictly Deza graphs were obtained in the article written srg by five authors. Lists of Deza graps and Cayley-Deza M. Erickson, S. Fernando, W.H. Haemers, W.H. Hardy, J. graphs Hemmeter, Deza graphs with Deza graphs: A generalization of strongly regular graph disconnected second J. Combin. Designs 1999, V. 7, P. 395-405 neighborhood

Introduction On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs V.V. Kabanov invited his postgraduate student Galina Ermakova to Deza graphs without investigate a class Deza graphs without triangles and a class Deza 3-cocliques graphs without 3-cocliques. Line graphs Deza graphs As the complement of strongly regular graph is also strongly regular obtained from srg graph, these questions for strongly regular graphs are the same. But Lists of Deza it is not true for Deza graphs. graps and Cayley-Deza graphs There are exactly seven triangle-free strongly regular graphs known: Deza graphs the five cycle, the Petersen Graph, the Clebsch Graph, the with Hoffman-Singleton Graph, the Gewirtz Graph, the Higman-Sims disconnected second Graph, and a (77 , 16 , 0 , 4) strongly regular subgraph of the neighborhood Higman-Sims graph. Every Moore Graph of diameter 2 is a triangle-free strongly regular graph, so if there is a 57-regular Moore Graph of diameter 2, this would add another to the list.

Moore graphs On some classes of Deza graphs A Moore graph is a regular graph of degree k and diameter d whose V.V. Kabanov, number of vertices equals to the upper bound L.V. Shalaginov Deza graphs d − 1 Deza graphs � ( k − 1) i . 1 + k without 3-cocliques i =0 Line graphs The Hoffman–Singleton theorem states that any Moore graph with Deza graphs girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are: obtained from srg The complete graphs K n on n > 2 vertices. (diameter 1, girth 3, Lists of Deza graps and degree n − 1 , order n ) Cayley-Deza graphs The odd cycles C 2 n +1 . (diameter n, girth 2 n + 1 , degree 2, order Deza graphs 2 n + 1 ) with disconnected The Petersen graph. (diameter 2, girth 5, degree 3, order 10) second neighborhood The Hoffman–Singleton graph. (diameter 2, girth 5, degree 7, order 50) A hypothetical graph of diameter 2, girth 5, degree 57 and order 3250; it is currently unknown whether such a graph exists. Unlike all other Moore graphs, Higman proved that the unknown Moore graph cannot be vertex-transitive. Machaj and Shiran and further proved that the order of the automorphism group of such a graph is at most 375.

Example On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov It is easy to see n -cube graph for n > 2 has the diameter n , girth 4, degree n and order 2 n . So it is a Deza graph with parameters Deza graphs (2 n , n, 0 , 2) . Deza graphs These Deza graphs do not have triangles. Note, that the complement without 3-cocliques of n -cube is a strictly Deza graph without 3 -cocliques with Line graphs parameters (2 n , 2 n − n − 1 , 2 n − 2 n, 2 n − 2 n − 2) . Deza graphs It’s clear Deza graphs without 3-cocliques are coedge regular graphs. obtained from srg So if Γ such Deza graph with parameters ( v, k, b, a ) , then Lists of Deza µ (Γ) ∈ { a, b } . graps and Cayley-Deza graphs Deza graphs with disconnected second neighborhood

Graphs without 3-cocliques On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Ermakova proved if Deza graph with parameters ( v, k, b, a ) without 3 -cocliques that in case µ (Γ) = b we have b ∈ { a + 1 , a + 2 } . In the Deza graphs case b = a + 2 the complement of Γ is an amply regular graph with Deza graphs without parameters ( v, v − k − 1 , 0 , 2) . 3-cocliques Line graphs Let v − k − 1 = l. It is interesting amply regular graphs with Deza graphs parameters ( v, l, 0 , 2) were investigated before by Andries E. obtained from srg Brouwer for degree l less than 8 . Lists of Deza graps and Andries E. Brouwer. Classification of small (0 , 2) -graphs Journal of Cayley-Deza graphs Combinatorial Theory, Series A 113 (2006) 1636–1645 Deza graphs www.elsevier.com/locate/jcta with disconnected second Andries E. Brouwer, P. R. J. Ostergard find the 302 graphs of degree neighborhood 8. It is also known amply regular graph with parameters ( v, l, 0 , 2) whose diameter equals to valency is n -cube. On this conference we have abstract of Ahkhamova about Deza graphs without 3-coclique with µ (Γ) = a where 1 ≤ a ≤ 3 .

Definition On some classes of Deza graphs V.V. Kabanov, L.V. Shalaginov Deza graphs For a given graph Γ , its line graph L (Γ) is the graph which vertices Deza graphs without are edges of the graph Γ , and two vertices are adjacent if and only if 3-cocliques the corresponding edges have exactly one common vertex in Γ . Line graphs Deza graphs obtained from srg Lists of Deza graps and Cayley-Deza graphs Deza graphs with disconnected second neighborhood

Lattice graph On some classes of Deza graphs V.V. Kabanov, For a positive integer n , the lattice graph L ( n ) is the graph with L.V. Shalaginov vertex set { 1 , . . . n } 2 in which vertex ( a, b ) is connected to vertex ( c, d ) if a = c or b = d . Thus, the vertices may be arranged at the Deza graphs points in an n × n -grid, with vertices being connected if they lie in Deza graphs without the same row or column. Alternatively, we can understand this 3-cocliques graph as the line graph of a bipartite complete graph between two Line graphs sets of n vertices. It is routine to see that the parameters of this Deza graphs obtained from graph are: v = n 2 , k = 2( n − 1) , λ = n − 2 , µ = 2 . srg Lists of Deza graps and Cayley-Deza graphs Deza graphs with disconnected second neighborhood