## Some relations between distance-based polynomials of trees.(English)Zbl 1120.05304

Let $$\delta_x$$ be the degree of the vertex $$x$$ and let $$d(x,y)$$ denote the distance between vertices $$x$$ and $$y$$ in a graph $$G$$. The author studies the polynomials $$\sum_{\{x,y\}}\lambda^{d(x,y)}$$, $$\sum_{\{x,y\}}(\delta_x+\delta_y)\lambda^{d(x,y)}$$ $$\sum_{\{x,y\}}\delta_x\delta_y\lambda^{d(x,y)}$$, where the summation goes over all sets $$\{x,y\}$$. Some relations between these polynomials are established in the case when $$G$$ is a tree.

### MSC:

 05C12 Distance in graphs

### Keywords:

distance in graphs; vertex degree
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