Equitable partitions of Latin‐square graphs

Abstract

We study equitable partitions of Latin‐square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue −3.

Perfect 2‐colorings of Hamming graphs

“…A very nice result of Potapov [41] shows a one‐to‐one correspondence between the perfect 2‐colorings of the Hamming graph and the Boolean‐valued functions on attaining a bound that connects the correlation immunity of the function, the density of ones, and the average 0–1‐contact number (the number of neighbors with function value 1 for a given vertex with value 0). Besides the Hamming graphs, distance‐regular graphs where perfect colorings have been studied include Johnson graphs, see, for example, [1,16], Latin‐square graphs [2], halved hypercubes [32], Grassmann graphs , see, for example, [10,37]. In finite geometry, perfect 2‐colorings are studied as intriguing sets, see, for example, [3].…”

Subgroup regular sets in Cayley graphs

“…Now assume that Γ is a connected k-regular graph with vertex set V . Then k is a simple eigenvalue of M [14, Theorem 9.3.3], and the equitable partition V of Γ is said to be µ-equitable [1] if all eigenvalues of M other than k are equal to µ. In particular, if an equitable partition {C, V \ C} is µ-equitable, then the nonempty proper subset C of V is called a µ-perfect set [1].…”

“…See, for example, [12] for a study of perfect 2-colorings of Johnson graphs J(v, 3), and [3,23] for some recent results on perfect 2-colourings of Hamming graphs. In [1], equitable partitions of Latin square graphs are studied and those whose quotient matrix does not have an eigenvalue −3 are classified. In [2], a few results on equitable partitions and regular sets of Cayley graphs involving irreducible characters of the underlying groups are obtained.…”

“…In [1], equitable partitions of Latin square graphs are studied and those whose quotient matrix does not have an eigenvalue −3 are classified. In [2], a few results on equitable partitions and regular sets of Cayley graphs involving irreducible characters of the underlying groups are obtained.…”

“…In [18, Theorem 2.2], it was proved that, for a normal subgroup H of a group G, H is a perfect code of G if and only if (1) for any g ∈ G with g 2 ∈ H, there exists h ∈ H such that (gh…”