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

Abstract

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 $K{f}^{*}(G)$ 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 $\frac{K{f}^{*}(G)}{\mathit{Gut}(G)}$ of spiro and polyphenyl hexagonal chains, respectively.

Keywords:

• Degree-Kirchhoff index
• Gutman index
• resistance distance
• distance
• spiro hexagonal chain
• polyphenyl hexagonal chain

Acknowledgments

The authors would like to thank editor and the anonymous referees for their valuable comments and helpful suggestions for the improvement of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Natural Science Foundation of Xinjiang Province [No. 2021D01C069], the Scientific Research Plan of Universities in Xinjiang, China [No. XJEDU2021I001] and the National Natural Science Foundation of China [No. 12161085].