TY - JOUR

T1 - Integral Cayley Graphs

AU - Guo, W.

AU - Lytkina, D. V.

AU - Mazurov, V. D.

AU - Revin, D. O.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - Let G be a group and S ⊆ G a subset such that S = S−1, where S−1 = {s−1 | s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) | g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ sk ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook.

AB - Let G be a group and S ⊆ G a subset such that S = S−1, where S−1 = {s−1 | s ∈ S}. Then a Cayley graph Cay(G, S) is an undirected graph Γ with vertex set V (Γ) = G and edge set E(Γ) = {(g, gs) | g ∈ G, s ∈ S}. For a normal subset S of a finite group G such that s ∈ S ⇒ sk ∈ S for every k ∈ ℤ which is coprime to the order of s, we prove that all eigenvalues of the adjacency matrix of Cay(G, S) are integers. Using this fact, we give affirmative answers to Questions 19.50(a) and 19.50(b) in the Kourovka Notebook.

KW - adjacency matrix of graph

KW - Cayley graph

KW - character of group

KW - complex group algebra

KW - integral graph

KW - spectrum of graph

UR - http://www.scopus.com/inward/record.url?scp=85075253805&partnerID=8YFLogxK

U2 - 10.1007/s10469-019-09550-2

DO - 10.1007/s10469-019-09550-2

M3 - Article

AN - SCOPUS:85075253805

VL - 58

SP - 297

EP - 305

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 4

ER -