Degree resistance distance of trees with some given parameters

Document Type: Research Paper

Authors

1 College of Mathematics and Physics, Huanggang Normal University, Huanggang, China

2 College of Life Science and Techonolgy, Huazhong University of Science and Technology, Wuhan

Abstract

The degree resistance distance of a graph $G$ is defined as $D_R(G)=\sum_{i<j}(d(v_i)+d(v_j))R(v_i,v_j)$, where $d(v_i)$ is the degree of the vertex $v_i$, and $R(v_i,v_j)$ is the resistance distance between the vertices $v_i$ and $v_j$. Here we characterize the extremal graphs with respect to degree resistance distance among trees with given diameter, number of pendent vertices, independence number, covering number, and maximum degree, respectively.

Keywords

Main Subjects


[1] P. Ali, S. Mukwembi and S. Munyira, Degree distance and vertex{connectivity, Discrete Appl. Math. , 161 (2013)
2802{2811.
[2] C. Arauz, The Kirchhoff indexes of some comp osite networks, Discrete Appl. Math. , 160 (2012) 1429{1440.
[3] B. Bollobas, Modern Graph Theory , Springer-Verlag, 1998.
[4] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications , American Elsevier Publishing Co., Inc., New
York, 1976.
[5] S. B Chen, Q. Chen, X. Cai and Z. Guo, Maximal Degree Resistance Distance of Unicyclic Graphs, MATCH
Commun. Math. Comput. Chem. , 75 (2016) 157{168.
[6] A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. ,
66 (2001) 211{249.
[7] A. A. Dobrynin, I. Gutman, S. Klavzar and P. Zigert, Wiener index of hexagonal systems, Acta Appl. Math. , 72
(2002) 247{294.
[8] A. A. Dobrynin and A. A. Ko chetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. , 34 (1994) 1082{1086.
[9] P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, On the degree distance of a graph, Discrete Appl. Math., 157 (2009) 2773{2777.
[10] J. F. Du, G. F. Su, J. H. Tu and I. Gutman, The degree resistance distance of cacti, Discrete Appl. Math. , 188
(2015) 16{24.
[11] Z. B. Du and B. Zhou, The Estrada index of trees, Linear Algebra Appl. , 435 (2011) 2462{2467.
[12] L. Feng, G. Yu, K. Xu and Z. Jiang, A note on the Kirchhoff index of bicyclic graphs, Ars Comb. , 114 (2014) 33{40.
[13] I. Gutman, J. Rada and O. Araujo, The Wiener index of starlike trees and a related partial order, MATCH Commun. Math. Comput. Chem. , 42 (2000) 145{154.
[14] I. Gutman, L. Feng and G. Yu, On the degree resistance distance of unicyclic graphs, Trans. Comb. , 1 no. 2 (2012) 27{40.
[15] Q. Guo, H. Deng and D. Chen, The extremal Kirchhoff index of a class of unicyclic graphs, MATCH Commun.
Math. Comput. Chem. , 61 (2009) 713{722.
[16] F. Huang, X. l. Li and S. j. Wang, On Maximum Laplacian Estrada Indices of Trees with Some Given Parameters,
MATCH Commun. Math. Comput. Chem. , 74 (2015) 419{429.
[17] F. Huang, X. l. Li and S. j. Wang, On maximum Estrada indices of bipartite graphs with some given parameters,
Linear Algebra Appl. , 465 (2015) 283{295.
[18] A. Ilic, S. Klavzar and D. Stevanovic, Calculating the degree distance of partial hamming graphs, MATCH Commun. Math. Comput. Chem. , 63 (2010) 411{424.
[19] A. Ilic, A Note on the Additive Degree Kirchhoff Index, MATCH Commun. Math. Comput. Chem. , 75 (2016)
223{226.
[20] D. J. Klein, Resistance-distance sum rules, Croat. Chem. Acta. , 75 (2002) 633{649.
[21] D. J. Klein and M. Randic, Resistance distance, J. Math. Chem. , 12 (1993) 81{95.
[22] S. C. Li, Y. B. Song and H. H. Zhang, On the degree distance of unicyclic graphs with given matching numb er,
Graphs Combin. , 31 (2015) 2261{2274.
[23] J. B. Liu et al. On degree resistance distance of cacti, Discrete Appl. Math. , 203 (2016) 217{225.
[24] M. H. Liu and B. L. Liu, A Survey on Recent Results of Variable Wiener Index, MATCH Commun. Math. Comput.
Chem. , 69 (2013) 491{520.
[25] J. L. Palacios, Upp er and lower b ounds for the additive degreeCKirchhoff index, MATCH Commun. Math. Comput. Chem. , 70 (2013) 651{655.
[26] I. Tomescu, Prop erties of connected graphs having minimum degree distance, Discrete Math. , 309 (2009) 2745{2748.
[27] I. Tomescu, Ordering connected graphs having small degree distances, Discrete Appl. Math. , 158 (2010) 1714{1717.
[28] J. H. Tu, J. f. Du and G. f. Su, The unicyclic graphs with maximum degree resistance distance, Appl. Math. Comput., 268 (2015) 859{864.
[29] H. Wiener, Structural determination of paraffin b oiling p oints, J. Am. Chem. Soc. , 69 (1947) 17{20.
[30] W. H. Wang, Estrada Indices of the Trees with a Perfect Matching, MATCH Commun. Math. Comput. Chem. , 75
(2016) 373{383.
[31] Y. Yang, X. Jiang, Unicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. , 60
(2008) 107{120.
[32] Y. Yang, D.J. Klein, Resistance distance-based graph invariants of sub divisions and triangulations of graphs, Discrete Appl. Math. , 181 (2015) 260{274.
[33] B. Zhou, N. Trina jstic, On resistance-distance and kirchhoff index, J. Math. Chem. , 46 (2009) 283{289.
[34] H. Zhang, X. Jiang, Y. Yang, Bicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput.
Chem. , 61 (2009) 697{712.