Some new results on distance-based polynomials (Q2853257)

For a finite graph \(G\), Hosoya polynomials are \(H(G,x)=\sum\{x^{d(u,v)}\mid u,v\in V(G)\}\), \(H_e(G,x)=\sum\{x^{d(e,f)}\mid e,f\in E(G)\}\), and Gutman polynomials are \(H_1(G,x)=\sum\{(\deg(u)+\deg(v))x^{d(u,v)}\mid u,v\in V(G)\}\), \(H_2(G,x)=\sum\{(\deg(u) \deg(v))x^{d(u,v)}\mid u,v\in V(G)\}\), where \(x\) is a variable, \(d(u,v)\) is the distance of \(u\) and \(v\) in \(G\) for \(u,v\in V(G)\), \(d(A,B)=\min\{d(u,v)\mid u\in A,\;v\in B\}\) for \(A,B\subseteq V(G)\), \(\deg(u)\) is the degree of \(u\in V(G)\). The authors present basic relations between these polynomials and the form of these polynomials for special graphs and for some graph operations. Further the authors define \(W(G,x,y)=\sum \{x^{\deg(u)+\deg(v)}y^{d(u,v)}\mid u,v\in V(G)\}\), where \(x\) and \(y\) are variables, and they derive the form of \(W\) for the Cartesian product.