Let p be a prime number. Reducible cyclic codes of rank 2 over $$\mathbb {Z}_{p^m}$$ are shown to have exactly two Hamming weights in some cases. Their weight distribution is computed explicitly. When these projective, the coset graphs their dual strongly regular. The spectra determined.

We prove that if G and H are primitive strongly regular graphs with the same parameters and φ is a homomorphism from G to H, then φ is either an isomorphism or a coloring (homomorphism to a complete subgraph). Moreover, any such coloring is optimal for G and its image is a maximum clique of H. Therefore, the only endomorphisms of a primitive strongly regular graph are automorphisms or colorings...

A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte's bound (also called Hoffman's bound). Such spreads give rise to colorings meeting Hoffman's lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (part...

We consider strongly regular graphs r = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman-Singleton graph, and the triangular graphs T(q) with q = 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We...

We consider strongly regular graphs defined on a finite field by taking the union of some cyclotomic classes as difference set. Several new examples are found.

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...

In this paper, we examine the structure of vertexand edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parame...

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 se...

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m for every positive integer m. If m > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m, 2m +m, m +m, m +m). Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by ...