On highly regular strongly regular graphs

In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using our theoretical results we show that a family of non rank 3 graphs known to satisfy the $7$-vertex condition fulfills an even stronger condition, $(3,7)$-regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank $3$ strongly regular graphs satisfying the $6$-vertex condition. This strengthens and generalizes previous results by Reichard.Comment: 29 page

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Linear codes with complementary duals from some strongly regular subgraphs of the McLaughlin graph

We describe a number of properties of some ternary linearcodes defined by the adjacency matrices of some stronglyregular graphs that occur as induced subgraphs of the McLaughlin graph, namely the graphs withparameters $(105,72,51,45), (120,77,52,44), (176, 105, 68, 54),$ and$(253, 140, 87, 65)$ respectively. We show that the codes withparameters $[120,21,30]_3$,$[120,99,6]_3$, $[176, 21, 56]_3$, $[176, 155, 6]_3$, $[253, 22, 97]_3$ and $[253, 231, 8]_3$ obtained from these graphs are linear codes with complementary duals and thus meet the asymptotic Gilbert–Varshamov bound

Co-cliques and star complements in extremal strongly regular graphs

Suppose that the positive integer μ is the eigenvalue of largest multiplicity in an extremal strongly regular graph G. By interlacing, the independence number of G is at most 4μ2 + 4μ - 2. Star complements are used to show that if this bound is attained then either (a) μ = 1 and G is the Schläfli graph or (b) μ = 2 and G is the McLaughlin graph