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.


05C12 Distance in graphs
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