# Ordering of trees by multiplicative second Zagreb index

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Science , Faculty of Khansar, Khansar, Iran

2 Department of Mathematics and Computer Science, Faculty of Khansar, University of Isfahan, P.O.Box 87931133111, Khansar, Iran

Abstract

‎For a graph $G$ with edge set $E(G)$‎, ‎the multiplicative second Zagreb index of $G$ is defined as‎ ‎$\Pi_2(G)=\Pi_{uv\in E(G)}[d_G(u)d_G(v)]$‎, ‎where $d_G(v)$ is the degree of vertex $v$ in $G$‎.
‎In this paper‎, ‎we identify the eighth class of trees‎, ‎with the first through eighth smallest multiplicative second Zagreb indeces among all trees of order $n\geq 14$‎.

Keywords

Main Subjects

#### References

[1] J. Braun, A. Kerb er, M. Meringer and C. Rucker, Similarity of molecular descriptors: the equivalence of Zagreb indices and walk counts, MATCH Commun. Math. Comput. Chem., 54 (2005) 163-176.

[2] D. Vukievi, S. M. Ra jtma jer and N. Trina jsti, Trees with maximal second Zagreb index and prescrib ed numb er of vertices of the given degree, MATCH Commun. Math. Comput. Chem., 60 (2008) 65-70.

[3] K. C. Kinkar, A. Yurttas, M. Togan, A. S. Cevik and I. N. Cangu, The multiplicative Zagreb indices of graph
op erations, J. Inequal. Appl., 90 (2013) 1-14.

[4] M. Eliasi, A simple approach to orther the multiplicative Zagreb indices of connected graphs,Trans. Comb., 1 no. 4 (2012) 17-24.

[5] M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of rst Zagreb index, MATCH Commun. Math. Comput. Chem., 68 (2012) 217-230.

[6] I. Gutman, Multiplicative Zagreb indices of trees, Bul l. Int. Math. Virtual Inst., 1 (2011) 13-19.

[7] I. Gutman and N. Trina jstic, Graph theory and molecular orbitals Total p-electron energy of alternant hydro carb ons, Chem. Phys. Lett., 17 (1972) 535-538.

[8] J. Liu and Q. Zhang, Sharp upp er b ounds for multiplicative Zagreb indices, MATCH Commun. Math. Comput. Chem., 68 (2012) 231-240.

[9] S. Nikolic, G. Kovacevic, A. Milicevic and N. Trina jstic, The Zagreb indices 30 years after, Croat. Chem. Acta, 76 (2003) 113-124.

[10] T. Reti and I. Gutman, Relation b etween ordinary and multiplicative zagreb indices, Bul l. Int. Math. Virtual Inst., 2 (2012) 133-140.

[11] R. To deschini and V. Consonni, New lo cal vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem., 64 (2010) 359-372.

[12] R. To deschini, D. Ballabio and V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, in: I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors-Theory and Applications I, Univ. Kragujevac, Kragujevac, 2010 73-100.

[13] K. Xu and H. Hua, A uni ed approach to extremal multiplicative Zagreb indices for trees, Unicyclic and Bicyclic Graphs, MATCH Commun. Math. Comput. Chem., 68 (2012) 241-256.

[14] Z. Yan, Huiqing Liu and Heguo Liu, Sharp b ounds for the second Zagreb index of unicyclic graphs, J. Math. Chem., 42 (2007) 565-574.

[15] B. Zhou and I. Gutman, Relations b etween Wiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett., 394 (2004) 93-95.

[16] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem., 52 (2004) 113-118.

[17] B. Zhou and I. Gutman, Further prop erties of Zagreb indices, MATCH Commun. Math. Comput. Chem., 54 (2005) 233-239.