Algebraic degrees of quasi-abelian semi-Cayley digraphs (Q6611730)

Given a digraph \(\Gamma\), the splitting field \(F\) of \(\Gamma\) is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of \(\Gamma\). By the algebraic degree of \(\Gamma\) we denote the extension degree of \(F\) over the field of rational numbers \(\mathbb Q\).\N\NThis paper determines the splitting field and the algebraic degree of a special class of graphs. Namely, of quasi-abelian semi-Cayley digraphs over any finite group. A Cayley digraph Cay\((G, S)\) over a finite group \(G\) with respect to \(S\) is a digraph with vertex set \(G\) and edge set \(\{(x, y) | x, y \in G, yx^{-1}\in S\}\). A digraph is said to be a semi-Cayley digraph over a finite group \(G\) if it admits \(G\) as a semiregular automorphism group with two orbits of equal size. Specifically, for four subsets \(T_{ij} , i, j \in\{1, 2\}\), of a finite group \(G\), a semi-Cayley digraph \(SC(G, T_{11}, T_{22}, T_{12}, T_{21})\) is a digraph with vertex set the union of the right part \(G_1 = \{g_1 | g\in G\}\) and the left part \(G_2 = \{g_2 | g\in G\}\), and the arc set the union of \(\{(h_1, g_1) | gh^{-1}\in T_{11}\}\), \(\{(h_2, g_2) | gh^{-1}\in T_{22}\}\), \(\{(h_1, g_2) | gh^{-1}\in T_{12}\}\) and \(\{(h_2, g_1) | gh^{-1}\in T_{21}\}\). A semi-Cayley digraph \(SC(G, T_{11}, T_{22}, T_{12}, T_{21})\) is said to be quasi-abelian if each of \(T_{11}\), \(T_{22}\), \(T_{12}\) and \(T_{21}\) is a union of conjugacy classes of \(G\).\NIn particular, the algebraic degree of \(\Gamma\) is \(deg(\Gamma) = \frac{\varphi(m)|M|}{|T|}\), where \(\varphi\) is an Euler function, \(M =\langle [d_\chi(\chi(I_2 \setminus I_3)-\chi(I_3 \setminus I_2))]_K | \chi\in \text{Irr}(G)\rangle\),\N \(T = \{t\in\mathbb Z^\ast_m | (I_1)^t = I_1,(I_2 \setminus I_3)^t = I_2 \setminus I_3,(I_3 \setminus I_2)^t = I_3 \setminus I_2\}\) \Nand we assume that \(I_1 := T_{11}\cup T_{22}\), \(I_2 := (T_{11}T_{11})\cup (T_{22}T_{22})\cup (4\ast T_{12}T_{21})\), and \(I_3 := 2\ast T_{11}T_{22}\).