In an attempt to find a strongly regular graph of parameters (99, 14, 1, 2) or to disprove its existence, we studied its possible substructure and constructions.

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

The Friendship theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let λ be any nonnegative integer and μ be any positive integer. Suppose each pair of friends have exactly λ common friends and each pair of strangers have exactly μ common friends in a pa...

A subset of the vertex set of a graph G, S ⊆ V (G), is a (k, τ)-regular set if it induces a k-regular subgraph of G and every vertex not in the subset has τ neighbors in it. This paper is a contribution to the given problem of existence of (k, τ)-regular sets associated with all distinct eigenvalues of integral strongly regular graphs. The minimal idempotents of the Bose-Mesner algebra of stron...

Definition of the strongly regular graphs on 77 and 100 vertices belonging to the simple groups M22 and HiS.

Given the algebra T of ternions (upper triangular 2× 2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T 2. This set of points can be represented as a set of planes in the projective space over F 6. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admit...

Derivations, Jordan derivations, as well as automorphisms and Jordan automorphisms of the algebra of triangular matrices and some class of their subalgebras have been the object of active research for a long time [1, 2, 5, 6, 9, 10]. A well-know result of Herstein [11] states that every Jordan isomorphism on a prime ring of characteristic different from 2 is either an isomorphism or an anti-iso...

In this paper, we first introduce a novel class of graphs, namely supergrid. Supergrid graphs include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs. The Hamiltonian cycle (path) problem on supergrid graphs can be applied to ...

A subset of the vertex set of a graph G, S ⊆ V (G), is a (k, τ)-regular set if it induces a k-regular subgraph of G and every vertex not in the subset has τ neighbors in it. This paper is a contribution to the given problem of existence of (k, τ)-regular sets associated with all distinct eigenvalues of integral strongly regular graphs. The minimal idempotents of the Bose-Mesner algebra of stron...

We survey the relationships between two-weight linear [n, k] codes over GF(q), projective (n, k, /»„ /»,) sets in PG(£— \,q), and certain strongly regular graphs. We also describe and tabulate essentially all the known examples.