Let T be the subgroup of the multiplicative group C^(×)consisting of all complex numbers z with|z|=1.A T-gain graph is a tripleΦ=(G,T,φ)(or short for(G,φ))consisting of a simple graph G=(V,E),as the underlying...Let T be the subgroup of the multiplicative group C^(×)consisting of all complex numbers z with|z|=1.A T-gain graph is a tripleΦ=(G,T,φ)(or short for(G,φ))consisting of a simple graph G=(V,E),as the underlying graph of(G,Φ),the circle group T and a gain functionΦ:→E→T such that φ(vivj)=φ(vjvi) for any adjacent vertices vi and vj.Let i+(G,φ)(resp.,i+(G))be the positive inertia index of(G,φ)(resp.,G).In this paper,we prove that-c(G)≤i+(G,φ)-i+(G)≤c(G),where c(G)is the cyclomatic number of G,and characterize all the corresponding extremal graphs.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11971474)the Natural Science Foundation of Shandong Province(Grant No.ZR2019BA016)。
文摘Let T be the subgroup of the multiplicative group C^(×)consisting of all complex numbers z with|z|=1.A T-gain graph is a tripleΦ=(G,T,φ)(or short for(G,φ))consisting of a simple graph G=(V,E),as the underlying graph of(G,Φ),the circle group T and a gain functionΦ:→E→T such that φ(vivj)=φ(vjvi) for any adjacent vertices vi and vj.Let i+(G,φ)(resp.,i+(G))be the positive inertia index of(G,φ)(resp.,G).In this paper,we prove that-c(G)≤i+(G,φ)-i+(G)≤c(G),where c(G)is the cyclomatic number of G,and characterize all the corresponding extremal graphs.
