Regular weights of full rank on strongly regular graphs
The theory of regular weights on coherent configurations was developed by D. G. Higman. Let \(U_4\) be the set of fourth roots of unity. With respect to a given set \(X\), define a weight with values in \(U_4\) as a function \(\omega: X^2 \to U_4\) with the properties \(\omega (x,x)=1\) and \(\omega (x,y) = \overline {\omega (y,x)}\) for all \(x,y\in X\). If \(\omega\) is a weight, then \(\delta \omega: X^3 \to U_4\) is defined by \(\delta \omega (x,y,z) = \omega (x,y) \overline {\omega(x,z)} \omega (y,z)\). Let \(\Gamma\) be a strongly regular graph with vertex set \(X\) and relations \(f_i\) \((0\leq i\leq 2)\). Given \(x\in X\), define \(f_i(x) = \{y \in X \mid (x,y) \in f_i\}\). The following parameters link the weight \(\omega\) and the strongly regular graph \(\Gamma\). For \(x,z\in X\) and \(\alpha \in U_4\), set \(\beta_{ij}^{\delta \omega} (x, z, \alpha) = |\{y\in f_i(x) \cap f_j(z) \mid \delta \omega (x,y,z) = \alpha\}|\). If for \((x,z) \in f_k\) the cardinal \(\beta^{\delta \omega}_{ij} (x,z, \alpha)\) is independent of the choice of \((x,z) \in f_k\), the weight \(\omega\) is called regular on \(\Gamma\). The rank of \(\omega\) is defined as the number of relations on which \(\omega\) does not vanish. In this paper are considered regular weights with values in \(\{-1,1\}\) of rank 3 on the triangular graph \(T(5)\), on rank 3 graphs for \(\text{PSp}_4(q)\), \(q\) odd, and on \(L_3(4)\) (in particular, the weight parameters \(\beta^k_{ij}\) are computed). The main result is the classification of regular weights with values in \(\{-1,1\}\) of rank 3 on the lattice graph \(L_2(n)\). Theorem. If \(\omega\) is a nontrivial regular weight with full support on the lattice graph \(L_2(n)\) then \(n\) is even and \(\omega = \omega_1 \otimes \omega_2\), where \(\delta \omega_1\) and \(\delta \omega_2\) are regular 2-graphs with the same parameters.
scientific article; zbMATH DE number 952556
scientific article