Ternary codes from the strongly regular (45, 12, 3, 3) graphs and orbit matrices of 2-(45, 12, 3) designs

The enumeration of strongly regular graphs with parameters (45, 12, 3, 3) has been completed, and it is known that there are 78 non-isomorphic strongly regular (45, 12, 3, 3) graphs. A strongly regular graph with these parameters is a symmetric (45, 12, 3) design having a polarity with no absolute points. In this paper we examine the ternary codes obtained from the adjacency (resp. incidence) matrices of these graphs (resp. designs), and those of their corresponding derived and residual designs. Further, we give a generalization of a result of Harada and Tonchev on the construction of non-binary self-orthogonal codes from orbit matrices of block designs under an action of a fixed-point-free automorphism of prime order. Using the generalized result we present a complete classification of self-orthogonal ternary codes of lengths 12, 13, 14, and 15, obtained from non-fixed parts of orbit matrices of symmetric (45, 12, 3) designs admitting an automorphism of order 3. Several of the codes obtained are optimal or near optimal for the given length and dimension. We show in addition that the dual codes of the strongly regular (45, 12, 3, 3) graphs admit majority logic decoding.     

The Search for Pseudo Orthogonal Latin Squares of Order Six

We report on the completecomputer search for a strongly regular graph with parameters(36,15,6,6) and chromatic number six. The resultis that no such graph exists.     

Strongly Regular Decompositions of the Complete Graph

We study several questions about amorphic association schemes and other strongly regular decompositions of the complete graph. We investigate how two commuting edge-disjoint strongly regular graphs interact. We show that any decomposition of the complete graph into three strongly regular graphs must be an amorphic association scheme. Likewise we show that any decomposition of the complete graph into strongly regular graphs of (negative) Latin square type is an amorphic association scheme. We study strongly regular decompositions of the complete graph consisting of four graphs, and find a primitive counterexample to A.V. Ivanov's conjecture which states that any association scheme consisting of strongly regular graphs only must be amorphic.     

GQ(4, 2)-extensions, strongly regular case

A strongly regular locallyGQ(4, 2)-graph is a graph with parameters either (126, 45, 12, 8) or (190, 45, 12, 10). The existence and the uniqueness of the corresponding locallyGQ(4, 2)-graph in the first case are well known. We prove that theGQ(4, 2)-hyperoval on ten vertices either is the Petersen graph, or is the Möbius 5-prism, or consists of two (2, 3)-subgraphs connected by three edges. We obtain homogeneousGQ(4, 2)-solutions with a strongly regular point graph; in particular, this implies the negative answer to the question of F. Buekenhout concerning the existence of a locallyGQ(4, 2)-graph with the parameters (190, 45, 12, 10).     

On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)

We prove that a strongly regular graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric graph for pG ₂(5, 32), we conclude that the partial geometries pG ₂(5, 32) and pG ₂(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric graph for pG ₃(6, 80) is a pseudogeometric graph for pG ₂(5, 32) and, therefore, a pseudogeometric graph for the partial geometry pG ₃(6, 80) [i.e., a strongly regular graph with parameters (1127, 486, 165, 243)] does not exist.     

Spreads in Strongly Regular Graphs

A spread of a strongly regular graph is a partitionof the vertex set into cliques that meet Delsarte's bound (alsocalled Hoffman's bound). Such spreads give rise to coloringsmeeting Hoffman's lower bound for the chromatic number and tocertain imprimitive three-class association schemes. These correspondenceslead to conditions for existence. Most examples come from spreadsand fans in (partial) geometries. We give other examples, includinga spread in the McLaughlin graph. For strongly regular graphsrelated to regular two-graphs, spreads give lower bounds forthe number of non-isomorphic strongly regular graphs in the switchingclass of the regular two-graph.     

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.     

A Classification of the Strongly Regular Generalised Johnson Graphs

The generalised Johnson graphs are the graphs J(n, k, m) whose vertices are the k subsets of {1, 2, . . . , n}, with two vertices J ₁ and J ₂ joined by an edge if and only if ${{|J_1 \cap J_2| = m}}$ . A graph is called d-regular if every vertex has exactly d edges incident to it. A d-regular graph on v vertices is called a (v, d, a, c)-strongly regular graph if every pair of adjacent vertices have exactly a common neighbours and every pair of non-adjacent vertices have exactly c common neighbours. The triangular graphs J(n, 2, 1), their complements J(n, 2, 0), the sporadic examples J(10, 3, 1) and J(7, 3, 1), as well as the trivially strongly regular graphs J(2k, k, 0) are examples of strongly regular generalised Johnson graphs. In this paper we prove that there are no other strongly regular generalised Johnson graphs.     

On strongly regular self - complementary graphs

It is shown that certain conditions assumed on a regular self-complementary graph are not sufficient for the graph to be strongly regular, answering in the negative a question posed by Kotzig in [1].     

On the partial geometry pg(6, 6, 2)

We give some new representations of the partial geometry pg(6, 6, 2), which was constructed by van Lint and Schrijver, show a connection with a strongly regular graph based on the ternary Golay code, and determine the automorphism group of the geometry.     

Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

On betweenness-uniform graphs

The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distanceregular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for sufficiently large n, there are superpolynomially many betweenness-uniform graphs on n vertices, and explore the structure of betweenness-uniform graphs having a universal or sub-universal vertex.     

A geometrical characterization of strongly regular graphs

We give a necessary and sufficient condition of a Euclidean representation of a simple graph to be spherical. Moreover we show a characterization of strongly regular graphs from the view point of Euclidean representations of a graph. From this characterization, we define a natural generalized concept of a strongly regular graph closely related to Euclidean designs and codes.     

A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs

In this paper we study a distance-regular graph Γ of diameter d ≥? 3 which satisfies the following two conditions: (a) For any integer i with 1 ≤? i ≤? d ? 1 and for any pair of vertices at distance i in Γ there exists a strongly closed subgraph of diameter i containing them; (b) There exists a strongly closed subgraph Δ which is completely regular in Γ. It is known that if Δ has diameter 1, then Γ is a regular near polygon. We prove that if a strongly closed subgraph Δ of diameter j with 2 ≤? j ≤? d ? 1 is completely regular of covering radius d ? j in Γ, then Γ is either a Hamming graph or a dual polar graph.     

An Inextensible Association Scheme Associated with a 4-regular Graph

 The scheme associated with a graph is an association scheme iff the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme in the case of a colored, directed graph. If a presuperscheme associated with a graph is of height t, then the graph satisfies the (t+3)-vertex condition. On the other hand, the current paper provides an example of a regular and 3-regular graph, satisfying the 4-vertex condition, whose association scheme cannot be extended to a presuperscheme of height 1. Received: March 4, 1998 Final version received: March 15, 1999     

Links between Latin squares,nets, graphs and groups: Work inspired by a paper of A. Barlotti and K. Strambach

Starting from a computer generated example of a transitive Latin square graph over a proper loop we describe a computer free interpretation of this specific strongly regular graph. Moreover, with this interpretation we were able to generalize the result to an infinite series of similar examples.     

Strongly regular graphs and finite Ramsey theory

Some connections between strongly regular graphs and finite Ramsey theory are drawn. Let B_n denote the graph K₂+K?_n. If there exists a strongly regular graph with parameters (υ, k, λ, μ), then the Ramsey number r(B_λ+1, B_{υ?2k+μ ?1})?υ+1. We consider the implications of this inequality for both Ramsey theory and the theory of strongly regular graphs.     

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.