A characterization of the Hamming graph by strongly closed subgraphs

A Distance-Regular Graph with Strongly Closed Subgraphs

Let be a distance-regular graph of diameter d, valency k and r := max{i | (ci , bi ) = (c1, b1)}. Let q be an integer with r + 1 ≤ q ≤ d − 1. In this paper we prove the following results: Theorem 1 Suppose for any pair of vertices at distance q there exists a strongly closed subgraph of diameter q containing them. Then for any integer i with 1 ≤ i ≤ q and for any pair of vertices at distance i ...

Random Subgraphs of the 2d Hamming Graph: the Supercritical Phase

We study random subgraphs of the 2-dimensional Hamming graph H(2, n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p = 1+ε 2(n−1) for some ε ∈ R. In [4, 5], the size of the largest connected component was estimated precisely for a large class of graphs including H(2, n) for ε ≤ ΛV −1/3, where Λ > 0 is a constant and V = n denotes ...

Connectivity threshold for random subgraphs of the Hamming graph

We study the connectivity of random subgraphs of the d-dimensional Hamming graph H(d, n), which is the Cartesian product of d complete graphs on n vertices. We sample the random subgraph with an i.i.d. Bernoulli bond percolation onH(d, n) with parameter p. We identify the window of the transition: when np−logn → −∞ the probability that the graph is connected goes to 0, while when np − logn → +∞...

Classification of some strongly regular subgraphs of the McLaughlin graph

By means of an exhaustive computer search we have proved that the strongly regular graphs with parameters (v, k, λ, μ) = (105, 32, 4, 12), (120, 42, 8, 18) and (176, 70, 18, 34) are unique upto isomorphism. Each of these graphs occurs as an induced subgraph in the strongly regular McLaughlin graph. We have used an orderly backtracking algorithm with look-ahead and look-back strategies, applying...

The Maximum Number of Edges in a Strongly Multiplicative Graph