Kyungpook Mathematical Journal 2018; 58(2): 221-229
Published online June 23, 2018
Copyright © Kyungpook Mathematical Journal.
Bounds for the First Zagreb Eccentricity Index and First Zagreb Degree Eccentricity Index
P. Padmapriya* and Veena Mathad
Received: August 28, 2017; Accepted: June 8, 2018
The first Zagreb eccentricity index
Keywords: eccentricity, diameter, radius, Zagreb eccentricity indices, total eccentricity of a graph
A systematic study of topological indices is one of the most striking aspects in many branches of mathematics with its applications and various other fields of science and technology. A topological index is a numeric quantity from the structural graph of a molecule. According to the IUPAC definition, a topological index (or molecular structure descriptor) is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity.
All the graphs
The Zagreb indices were introduced by Gutman and Trinajsti
The invariants based on vertex eccentricities attracted some attention in Chemistry. In an analogy with the first and the second Zagreb indices, M. Ghorbani et al. and D. Vuki
The Zagreb degree eccentricity indices are introduced in . First Zagreb degree eccentricity index (
The total eccentricity index of G is defined as
In this paper we obtain some bounds for the first Zagreb eccentricity index and first Zagreb degree eccentricity index.
We can see the appearance of Theorem 2.1, in .
Since equality in (
Equality of (
Equality holds if and only if
Since equality in 2.4 holds if and only if
Equality of (
The Lagrange identity is as follows.
(i) If we set
By power-mean inequality , we have
with equality if and only if
for any 2 ≤
From the above, we get
with equality if and only if
Using the above result in (
Using the above result we get
(ii) If we set
using the above result in (
The equality holds in (
In this paper we have established some bounds of the first Zagreb eccentricity index and first Zagreb degree eccentricity index in terms of some graph parameters such as order, size, maximum and minimum degree, radius, diameter and total eccentricity index. It may be useful to give the bounds for
The first author is thankful to the University Grants Commission, Government of India, for the financial support under the Basic Science Research Fellowship. UGC vide No.F.25 – 1/2014 – 15(BSR)/7 – 349/2012(BSR), January 2015.
- Biernacki, M, Pidek, H, and Ryll-Nardzewsk, C (1950). Sur une inégalité entre des intégrales définies. Ann Univ Mariae Curie-Skodowska Sect A. 4, 1-4.
- Biler, P, and Witkowski, A (1990). Problems in Mathematical Analysis. New York: Marcel Dekker, Inc
- Bullen, PS, Mitrinović, DS, and Vasić, PM (1988). Means and their inequalities. Reidel: Dordrecht
- De, N, Nayeem, SMA, and Pal, A (2015). Total eccentricity index of the generalized hierarchical product of graphs. Int J Appl Comput Math. 1, 503-511.
- Diaz, JB, and Metcalf, FT (1963). Stronger forms of a class of inequalities of G. Pólya-G. Szegö and L. V. Kantorovich. Bull Amer Math Soc. 69, 415-418.
- Fathalikhani, K, Faramarzi, H, and Yousefi-Azari, H (2014). Total eccentricity of some graph operations. Electron Notes Discrete Math. 45, 125-131.
- Ghorbani, M, and Hosseinzadeh, MA (2012). A new version of Zagreb indices. Filomat. 26, 93-100.
- Gutman, I, and Trinajstić, N (1972). Graph theory and molecular orbitals Total π - electron energy of alternant hydrocarbons. Chem Phys Lett. 17, 535-538.
- Harary, F (1969). Graph theory. Reading Mass: Addison-Wesley
- Milovanovć, IŽ, Milovanovć, EI, and Zakić, A (2014). A short note on graph energy. MATCH Commun Math Comput Chem. 72, 179-182.
- Mitrinović, DS (1970). Analytic inequalities. Berlin-Heidelberg-New York: Springer-Verlag
- Nikoli, S, Kovačević, ćG, Milićević, A, and Trinajstić, N (2003). The Zagreb indices 30 years after. Croat Chem Acta. 76, 113-124.
- Padmapriya, P, and Mathad, V (). Zagreb degree eccentricity indices of graphs. Novi Sad J Math.
- Vukicevic, D, and Graovac, A (2010). Note on the comparison of the first and second normalized Zagreb eccentricity indices. Acta Chim Slov. 57, 524-528.
- Van de Waterbeemd, H, Carter, RE, Grassy, G, Kubiny, H, Martin, YC, Tutte, MS, and Willet, P (1997). Glossary of terms used in computational drug design. Pure Appl Chem. 69, 1137-1152.