Some Properties of a Class of Addition Cayley Signed Graph

Let G=Cay^(+)(Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk),Φ)be a simple graph having vertex set V(G)=Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk)and edge set E(G)={{x,y}:x+y∈Φ},where all p_(1),p_(2),…,p_(k)are distinct prime factors andΦis the set of all units of the ring Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk).LetΣ=(G,σ)be a signed graph whose underlying graph is G and signature function isσ:E(G)→{+1,-1}defined asσ({x,y})={+1,ifx∈φ(p_(1))or y∈φ(p_(1)^(α1)p_(2)^(α2)…p_(k)^(αk));-1,otherwise.In this paper,we characterize the balance ofΣand some graphs derived from it such asη(Σ),L(Σ)and C_(E)(Σ).Moreover,we investigate the clusterability and sign-compativility ofΣ.