General degree distance of graphs

Abstract

We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b \in \mathbb{R}$, the general degree distance $DD_{a,b} (G) = \sum_{ v \in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = \sum_{ w \in V(G) \setminus \{ v \} } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices.

Keywords

• Degree distance
• Chromatic number
• Vertex connectivity

References

1. [1] P. Ali, S. Mukwembi, S. Munyira, Degree distance and edge-connectivity, Australas. J. Combin. 60 (2014) 50–68.
2. [2] P. Ali, S. Mukwembi, S. Munyira, Degree distance and vertex-connectivity, Discrete Appl. Math. 161(18) (2013) 2802–2811.
3. [3] S. Chen, W. Liu and F., Xia, Extremal degree distance of bicyclic graphs, Util. Math. 90 (2013) 149–169.
4. [4] K. C. Das, G. Su, L. Xiong, Relation between degree distance and Gutman index of graphs, MATCH Commun. Math. Comput. Chem. 76(1) (2016) 221–232.
5. [5] A. A. Dobrynin, A. A. Kochetova, Degree distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34(5) (1994) 1082–1086.
6. [6] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34(5) (1994) 1087–1089.
7. [7] A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh, Minimum generalized degree distance of n-vertex tricyclic graphs, J. Inequal. Appl. 2013 (2013) 548.
8. [8] A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh, M. V. Diudea, Generalized degree distance of trees, unicyclic and bicyclic graphs, Stud. Univ. Babes-Bolyai Chem. 57(4) (2012) 73–85.

9. [9] H. Hua, H. Wang, X. Hu, On eccentric distance sum and degree distance of graphs, Discrete Appl. Math. 250 (2018) 262–275.
10. [10] S. Li, Y. Song, H. Zhang, On the degree distance of unicyclic graphs with given matching number, Graphs Combin. 31(6) (2015) 2261–2274.
11. [11] S. Li, H. Zhang, M. Zhang, Further results on the reciprocal degree distance of graphs, J. Comb. Optim. 31(2) (2016) 648–668.
12. [12] X. Li, J.-B. Liu, On the reciprocal degree distance of graphs with cut vertices or cut edges, Ars Combin. 130 (2017) 303–318.
13. [13] S. Mukwembi, S. Munyira, Degree distance and minimum degree, Bull. Aust. Math. Soc. 87(2) (2013) 255–271.
14. [14] K. Pattabiraman, P. Kandan, Generalized degree distance of strong product of graphs, Iran. J. Math. Sci. Inform. 10(2) (2015) 87–98.
15. [15] K. Pattabiraman, M. Vijayaragavan, Reciprocal degree distance of product graphs, Discrete Appl. Math. 179 (2014) 201–213.
16. [16] S. Sedghi, N. Shobe, Degree distance and Gutman index of two graph products, J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140.
17. [17] D. Sarala, S. K. Ayyaswamy, S. Balachandran, K. Kannan, A note on Steiner reciprocal degree distance, Discrete Math. Algorithms Appl. 12(4) (2020) 2050050.
18. [18] T. Vetrík, M. Masre, Generalized eccentric connectivity index of trees and unicyclic graphs, Discrete Appl. Math. 284 (2020) 301–315.
19. [19] H. Wang, L. Kang, Further properties on the degree distance of graphs, J. Comb. Optim. 31(1) (2016) 427–446.
20. [20] Z. Zhu, Y. Hong, Minimum degree distance among cacti with perfect matchings, Discrete Appl. Math. 205 (2016) 191–201.