Degree distance and Gutman index of increasing trees

Document Type: Research Paper

Authors

1 Department of statistics, Imam Khomeini International University, Qazvin

2 Imam Khomeini International University

Abstract

‎‎The Gutman index and degree distance of a connected graph $G$ are defined as‎
‎\begin{eqnarray*}‎
‎\textrm{Gut}(G)=\sum_{\{u,v\}\subseteq V(G)}d(u)d(v)d_G(u,v)‎,
‎\end{eqnarray*}‎
‎and‎
‎\begin{eqnarray*}‎
‎DD(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)+d(v))d_G(u,v)‎,
‎\end{eqnarray*}‎
‎respectively‎, ‎where‎ ‎$d(u)$ is the degree of vertex $u$ and $d_G(u,v)$ is the distance between vertices $u$ and $v$‎. ‎In this paper‎, ‎through a recurrence equation for the Wiener index‎, ‎we study the first two‎
‎moments of the Gutman index and degree distance of increasing‎ ‎trees‎.

Keywords

Main Subjects


[1] T. Ali Khan and R. Neininger, Tail bounds for the Wiener index of random trees, 2007 Conference on Analysis of Algorithms, AofA 07 (Discrete Math. Theoret. Comput. Sei. Proc. AH), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2007 279-289.

[2] V. Andova, D. Dimitrov, J. Fink and R. Skrekovski, Bounds on Gutman index, MATCH Commun. Math. Comput. Chem., 67 (2012) 515-524.

[3] S. B. Chen and W. J. Liu, Extremal modified Schultz index of bicyclic graphs, MATCH Commun. Math. Comput. Chem., 64 (2010) 767-782.

[4] S. B. Chen and W. J. Liu, Extremal unicyclic graphs with respect to modified Schultz index, Util. Math., 86 (2011) 87-94.

[5] L. H. Clark and J. W. Moon, On the general Randic index for certain families of trees, Ars Combin., 54 (2000) 223-235.

[6] P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, The edge-Wiener index of a graph, Discrete. Math., 309 (2009) 3452-3457.

[7] Q. Feng and Z. Hu, On the Zagreb index of random recursive trees, J. Appl. Probab., 48 (2011) 1189-1196.

[8] Q. Feng, H. M. Mahmoud and A. Panholzer, Limit laws for the Randic index of random binary tree models, Ann. Inst. Statist. Math., 60 (2008) 319-343.

[9] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci., 34 (1994) 1087-1089.

[10] B. Hollas, Asymptotically independent topological indices on random trees, J. Math. Chem., 38 (2005) 379-387.

[11] B. Hollas, The Covariance of Topological Indices that Depend on the Degree of a Vertex, MATCH Commun. Math. Comput. Chem., 54 (2005) 177-187.

[12] B. Hollas, On the Variance of Topological Indices that Depend on the Degree of a Vertex, MATCH Commun. Math. Comput. Chem., 54 (2005) 341-350.

[13] S. Janson, The Wiener index of simply generated random trees, Random Structures and Algorithms, 22 (2003) 337-358.

[14] S. Janson and P. Chassaing, The center of mass of the ISE and the Wiener index of trees, Electron. Comm. Probab., 9 (2004) 178-187.

[15] R. Kazemi, Probabilistic analysis of the first Zagreb index, Trans. Comb., 2 no. 2 (2013) 35-40.

[16] R. Kazemi, The second Zagreb index of molecular graphs with tree structure, MATCH Commun. Math. Comput. Chem., 72 (2014) 753-760.

[17] R. Kazemi, The eccentric connectivity index of bucket recursive trees, Iranian J. Math. Chem., 5 (2014) 77-83.

[18] R. Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl. Math., 198 (2016) 147-154.

[19] M. Kuba and A.Panholzer, On the distribution of distances between specified nodes in increasing trees, Discrete Appl. Math., 158 (2010) 489-506.

[20] S. Mukwembi, On the Upper Bound of Gutman Index of Graphs, MATCH Commun. Math. Comput. Chem., 68 (2012) 343-348.

[21] R. Neininger, The Wiener index of random trees, Combin. Probab. Comput., 11 (2002) 587-597.

[22] V. Sheeba Agnes, Degree distance and Gutman index of corona product of graphs, Trans. Comb., 4 no. 3 (2015) 11-23.

[23] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.

[24] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947) 17-20.