^{1}B913, Zhixin Building School of Mathematics Shandong University 27 Shandananlu Rd.

^{2}School of Mathematics, Shandong University

Abstract

Let $G$ be a connected graph. The multiplicative Zagreb eccentricity indices of $G$ are defined respectively as ${\bf \Pi}_1^*(G)=\prod_{v\in V(G)}\varepsilon_G^2(v)$ and ${\bf \Pi}_2^*(G)=\prod_{uv\in E(G)}\varepsilon_G(u)\varepsilon_G(v)$, where $\varepsilon_G(v)$ is the eccentricity of vertex $v$ in graph $G$ and $\varepsilon_G^2(v)=(\varepsilon_G(v))^2$. In this paper, we present some bounds of the multiplicative Zagreb eccentricity indices of Cartesian product graphs by means of some invariants of the factors and supply some exact expressions of ${\bf \Pi}_1^*$ and ${\bf \Pi}_2^*$ indices of some composite graphs, such as the join, disjunction, symmetric difference and composition of graphs, respectively.

K. C. Das, D. W. Lee and A. Graovac (2013). Some properties of the Zagreb eccentricity indices. Ars Math. Contemp.. 6, 117-125

2

N. De (2012). On multiplicative Zagreb eccenticity indices. South Asian J. Math.. 2 (6), 570-577

3

Z. Du, B. Zhou and N. Trinajstic (2012). Extremal properties of the Zagreb eccentricity indices. Croat. Chem. Acta. 85 (3), 359-362

4

M. Ghorbani and M. A. Hosseinzadeh (2012). A new version of Zagreb indices. Filomat. 26, 93-100

5

I. Gutman and N. Trinajstic (1972). Graph theory and molecular orbitals, Total pi electron energy of alternant hydrocarbons. Chem. Phys. Lett.. 17, 535-538

6

I. Gutman (2011). Multipicative Zagreb indices of trees. Bull. Internat. Math. Virt. Inst.. 1, 13-19

7

M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi (2009). The first and second Zagreb indices of some graph operations. Discrete Appl. Math.. 157, 804-811

8

R. Todeschini and V. Consonni (2010). New local vertex invariants and molecular descriptors based on functions of the vertex degree. MATCH Commun. Math. Comput. Chem.. 64, 359-372

9

D. Vukiucevic and A. Graovac (2010). Note on the comparison of the first and second normalized Zagreb eccentricity indices. Acta Chim. Slov.. 57, 524-528

10

D. B. West (1996). Introduction to Graph Theory. second ed., Prentice Hall, Inc., Upper Saddle River, NJ.

11

R. Xing, B. Zhou and N. Trinajstic (2011). On Zagreb eccentricity indices. Croat. Chem. Acta. 84 (4), 493-497