On eigenfunctions and maximal cliques of Paley graphs of square order

Abstract

In this paper we find new maximal cliques of size q+1 2 or q+3 2 , accordingly as q ≡ 1(4) or q ≡ 3(4), in Paley graphs of order q 2 , where q is an odd prime power. After that we use new cliques to define a family of eigenfunctions corresponding to both non-principal eigenvalues and having the cardinality of support q + 1, which is the minimum by the weight-distribution bound.

Minimum supports of functions on the Hamming graphs with spectral constraints

“…We note that g = a 2 (0, q − 1) · a 4 (0) + a 4 (q − 1) · a 2 (0, q − 1). Consequently g ∈ U [1,2] (2, q) due to Corollary 1. Thus g(x, y) has the minimum size of the support in U [1,2] (2, q) but g(x, y) ∈ F 2 (2, q, 1, 2) and g(y, x) ∈ F 2 (2, q, 1, 2).…”

“…Consequently g ∈ U [1,2] (2, q) due to Corollary 1. Thus g(x, y) has the minimum size of the support in U [1,2] (2, q) but g(x, y) ∈ F 2 (2, q, 1, 2) and g(y, x) ∈ F 2 (2, q, 1, 2). Similar function can be also constructed for arbitrary n > 2.…”

Minimum supports of eigenfunctions of graphs: a survey

“…The MS-problem was first formulated by Krotov and Vorob'ev [98] in 2014 (they considered the MS-problem for the Hamming graph). During the last six years, the MSproblem has been actively studied for various families of distance-regular graphs [8,44,62,64,88,89,91,92,93,98,96] and Cayley graphs on the symmetric group [51]. In particular, the MS-problem is completely solved for all eigenvalues of the Hamming graph [92,93] and asymptotically solved for all eigenvalues of the Johnson graph [96].…”

Minimum Supports of Eigenfunctions with the Second Largest Eigenvalue of the Star Graph

“…In more details, these connections are described in [13,14]. Problem 1 was studied for the bilinear forms graphs in [17], the cubical distance-regular graphs in [16], the Doob graphs in [3], the Grassmann graphs in [14], the Hamming graphs in [13,15,18,19,20,21], the Johnson graphs in [22] and the Paley graphs in [7]. We note that Problem 1 is completely solved for all eigenvalues only for the Hamming graph and for the Johnson graph.…”