To the theory of q-ary Steiner and other-type trades

Abstract

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k − 1, k, v) bitrades, extended 1-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-regular with certain parameters. As an application of the results, we find the minimum cardinality of q-ary Steiner T q (k − 1, k, v) bitrades and show a connection of minimum such bitrades with dual polar subgraphs of the Grassmann graph J q (v, k).

PI-eigenfunctions of the Star graphs

“…The study of eigenfunctions of graphs plays an important role in theoretical and applied research [4]. Eigenfunctions of graphs are related to various combinatorial structures such as perfect codes, equitable partitions, trades [10,11,14].…”

Minimum supports of functions on the Hamming graphs with spectral constraints

“…This problem is directly related to the problem of finding the minimum possible difference of two combinatorial objects and to the problem of finding the minimum cardinality of the trades. In more details, these connections are described in [6,7]. For more information about connections between trades and eigenfunctions see [4,5,6,7].…”

“…In more details, these connections are described in [6,7]. For more information about connections between trades and eigenfunctions see [4,5,6,7]. The problem of finding the minimum size of the support of eigenfunctions was studied for the Johnson graphs in [11], for the Doob graphs in [1], for the cubic distance-regular graphs in [9] and for the Paley graphs in [2].…”

On eigenfunctions and maximal cliques of Paley graphs of square order

“…Note that, given eigenvalue θ of a graph Γ, a vector consisting of values of an eigenfunction of Γ corresponding to the eigenvalue θ is an eigenvector of the adjacency matrix of this graph corresponding to the eigenvalue θ, where the values of the eigenfunction and the indexes of the adjacency matrix have matched ordering. There are several papers devoted to the extremal problem of studying graph eigenfunctions with minimum cardinality of support (for more details and motivation, see [6]). In [7], Valyuzhenich found the minimum cardinality of support of an eigenfunction corresponding to the largest non-principal eigenvalue of a Hamming graph H(n, q) and characterised such eigenfunctions with the minimum cardinality of support.…”

“…In [8], Vorob'ev, Mogilnykh and Valyuzhenich, for all eigenvalues of a Johnson graph J(n, ω), characterised eigenfunctions with minimum cardinality of support, where n is sufficiently large. In [6], the weightdistribution lower bound for cardinality of support of an eigenfunction of a distance-regular graph is discussed. It follows from [6, Corollary 1] that an eigenfunction of P (q 2 ) corresponding to the eigenvalue θ 2 = −1−q 2 has at least q + 1 non-zero values.…”