Let Γ = (X, R) denote a distance-regular graph with distance function ∂ and diameter d ≥ 3. For 2 ≤ i ≤ d, by a parallelogram of length i, we mean a 4-tuple xyzu of vertices in X such that ∂(x, y) = ∂(z, u) = 1, ∂(x, u) = i, and ∂(x, z) = ∂(y, z) = ∂(y, u) = i − 1. Suppose the intersection number a1 = 0, a2 6= 0 in Γ. We prove the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial and con...

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In this paper, we are giving MATLAB program to find the energy and Wiener index of unit graphs. Our program demonstrates an intrinsic relationship between the elements of ring and structural properties of graphs. The unit graph G(Zn) turns out to be strongly regular when n = 2k. Several new directions for further research are also indicated by means of raising problems.

Spreads of finite symplectic, orthogonal and unitary vector spaces are used to construct new strongly regular graphs having the same parameters as the perpendicularity graphs of the underlying vector spaces. Some of the graphs are. related to partial geometries, while others produce interesting symmetric designs.

In this paper we construct transitive t-designs from the linear groups L(2, q), q ≤ 23. Thereby we classify t-designs, t ≥ 2, admitting a transitive action of the linear groups L(2, q), q ≤ 23, up to 35 points and obtained numerous transitive designs, for 36 ≤ v ≤ 55. In many cases we proved the existence of t-designs with certain parameter sets. Among others we constructed t-designs with param...

A graph is called of type k if it is connected, regular, and has k distinct eigenvalues. For example graphs of type 2 are the complete graphs, while those of type 3 are the strongly regular graphs. We prove that for any positive integer n, every graph can be embedded in n cospectral, non-isomorphic graphs of type k for every k ≥ 3. Furthermore, in the case k ≥ 5 such a family of extensions can ...

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups Z 4 × Z4`−4k 2 for all k when ` is odd and for all k < ` when ` is even. Similarly, we construct partial difference sets wit...

Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [5] and [12]. It was shown that such graphs exist iff the corresponding group is isomorphic to Zpn ⊕Zpn , where p is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur rings method. As a consequence we obtain a complete classification of strongly...