Injective and Relative Injective Zagreb Indıces of Graphs

Yıl 2018, Cilt: 6 Sayı: 1, 117 - 127, 15.04.2018

Akram Alqesmah Anwar Alwardi R. Rangarajan

Öz

Let $G=(V,E)$ be a graph. The injective neighborhood of a vertex $u\in V(G)$ denoted by $N_{in}(u)$ is defined as $N_{in}(u)=\{v\in V(G):|\Gamma(u,v)|\geq 1\}$, where $|\Gamma(u,v)|$ is the number of common neighborhoods between the vertices $u$ and $v$ in $G$. The cardinality of $N_{in}(u)$ is called the injective degree of the vertex $u$ in $G$ and denoted by $deg_{in}(u)$, \cite{20}. In this paper, we introduce the injective Zagreb indices of a graph $G$ as $M_1^{inj}(G)=\sum_{u\in V(G)}\big[deg_{in}(u)\big]^2$, $M_2^{inj}(G)=\sum_{uv\in E(G)}deg_{in}(u)deg_{in}(v)$, respectively, and the relative injective Zagreb indices as $RM_1^{inj}(G)=\sum_{u\in V(G)}deg_{in}(u)deg(u)$, $RM_2^{inj}(G)=\sum_{uv\in E(G)}\big[deg_{in}(u)deg(v)+deg(u)deg_{in}(v)\big]$, respectively. Some properties of these topological indices are obtained. Exact values for some families of graphs and some graph operations are computed.

Anahtar Kelimeler

First injective Zagreb index, Second injective Zagreb index, First relative injective Zagreb index, Second relative injective Zagreb index

Kaynakça

• [1] A. Alwardi, B. Arsi´c, I. Gutman, N. D. Soner, The common neighborhood graph and its energy, Iran. J. Math. Sci. Inf. 7(2) (2012) 1-8.
• [2] Anwar Alwardi, R. Rangarajan and Akram Alqesmah, On the Injective domination of graphs, In communication.
• [3] A.R. Ashrafi, T. Doˇ sli ´ c, A. Hamzeha, The Zagreb coindices of graph operations, Discrete Applied Mathematics 158 (2010) 1571–1578.
• [4] J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of molecular descriptors: the equivalence of Zagreb indices and walk counts, MATCH Commun. Math. Comput. Chem. 54 (2005) 163–176.
• [5] T. Doˇ sli ´ c, Vertex-Weighted Wiener Polynomials for Composite Graphs, Ars Math. Contemp. 1 (2008) 66–80.
• [6] I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.
• [7] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals, Total p-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
• [8] F. Harary, Graph theory, Addison-Wesley, Reading Mass (1969).
• [9] M. H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Applied Mathematics 157 (2009) 804–811.
• [10] Modjtaba Ghorbani, Mohammad A. Hosseinzadeh, A new version of Zagreb indices, Filomat 26 (1) (2012) 93–100.
• [11] S. Nikoli ´ c, G. Kova ˇ cevi ´ c, A. Mili ˇ cevi ´ c, N. Trinajsti ´ c, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124.
• [12] Rundan Xing, Bo Zhou and Nenad Trinajstic, On Zagreb Eccentricity Indices, Croat. Chem. Acta 84 (4) (2011) 493—497.
• [13] B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 233–239.
• [14] B. Zhou, I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett. 394 (2004) 93–95.
• [15] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004) 113–118.