2022/12/22 by Dijian Wang, Yaoping Hou, Wang, Dijian +3 · 1 citation
Chemistry · Materials Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Magnetism in coordination complexes #Organometallic Complex Synthesis and Catalysis
paper · pdf · doi:10.48550/arxiv.2212.11460
openalex publication_date 2022/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the Turán problem of signed graphs version. 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 Gs,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 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∼ Gs,t. Moreover, e-(Gs,t)≤ \lfloor(n-2)/(2)\rfloor\lceil(n-2)/(2)\rceil+n-2, with equality holding if and only if Gs,t= GU\lfloor(n-2)/(2)\rfloor,\lceil(n-2)/(2)\rceil, where GU\lfloor(n-2)/(2)\rfloor,\lceil(n-2)/(2)\rceil is obtained from G\lfloor(n-2)/(2)\rfloor,\lceil(n-2)/(2)\rceil by switching at vertex set U=\v,u1,…,u\lfloor(n-2)/(2)\rfloor\. Secondly, we prove that if G is C3--free, then ρ(G)≤ (1)/(2)( √( n2-8)+n-4), with equality holding if and only if G∼ G1,n-3.