## Graphs of extremal weights.(English)Zbl 0963.05068

The authors consider the graph weight $$w_\alpha (G)=\sum _{e\in E(G)}w_{\alpha }(e)$$, where $$\alpha \neq 0$$ is fixed and $$w_{\alpha }(e)=w_{\alpha }(\{x,y\})=(d(x)d(y))^\alpha$$ with $$d(x)$$ being the degree of $$x$$. It is proved that for the Randić weight $$\alpha =-1/2$$ (in the first definition in the article the “$$-$$” is missing), if $$G$$ has order $$n$$ and no isolated vertex then $$w_{-1/2}(G)\geq \sqrt {n-1}$$, with equality only for stars. It is also proved that for every $$G$$ with $$m$$ edges one has $$w_{\alpha }(G)\leq m(((8m+1)^{1/2}-1)/2)^{2\alpha }$$ for $$0<\alpha \leq 1$$ and the opposite inequality for $$-1\leq \alpha <0$$, with equality attained iff $$G$$ is a complete graph plus some isolated vertices. Further results are given.

### MSC:

 05C35 Extremal problems in graph theory 05C07 Vertex degrees

### Keywords:

Randić weight; graph weight; degree