Extremal results for [formula omitted]-free signed graphs

Abstract

In this paper, we study a Turán-like problem in the context of signed graphs. Suppose that G˙ is a connected unbalanced signed graph of order n with e(G˙) edges and e−(G˙) negative edges, and let ρ(G˙) be the spectral radius of G˙. The signed graph G˙s,t (s+t=n−2) is obtained from an all-positive clique (Kn−2,+) with V(Kn−2)={u1,…,us,v1,…,vt} (s,t≥1) and two isolated vertices u and v by adding a negative edge uv and positive edges uu1,…,uus,vv1,…,vvt. Firstly, we prove that if G˙ is C3−-free, then e(G˙)≤n(n−1)2−(n−2), with equality holding if and only if G˙ is switching equivalent to G˙s,t. Secondly, we prove that if G˙ is C3−-free, then ρ(G˙)≤12(n2−8+n−4), with equality holding if and only if G˙ is switching equivalent to G˙1,n−3.