Degree-Kirchhoff Indices and Gutman Indices of Spiro and Polyphenyl Hexagonal Chains

It is well known that the spiro and the polyphenyl hexagonal chains are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Let Kf*(G) be the degree-Kirchhoff index and Gut(G) be the Gutman index of a connected graph G. In this paper, we first give the formulae for calculating the Gutman indices and the degree-Kirchhoff indices of spiro and polyphenyl hexagonal chains, respectively. Then, we describe a relationship between the Gutman (resp. degree-Kirchhoff) indices of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain, and obtain the extremal values and characterize the extremal graphs with respect to the Gutman (resp. degree-Kirchhoff) indices among all spiro and polyphenyl hexagonal chains with n hexagons, respectively. Finally, we determine the bounds for Kf*(G)Gut(G) of spiro and polyphenyl hexagonal chains, respectively.