Year 2018, Volume 6, Issue 1, Pages 117 - 127 2018-04-15

Injective and Relative Injective Zagreb Indıces of Graphs

Akram Alqesmah [1] , Anwar Alwardi [2] , R. Rangarajan [3]

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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.
First injective Zagreb index, Second injective Zagreb index, First relative injective Zagreb index, Second relative injective Zagreb index
  • [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.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Author: Akram Alqesmah (Primary Author)
Country: India


Author: Anwar Alwardi
Country: Yemen


Author: R. Rangarajan
Country: India


Dates

Publication Date: April 15, 2018

Bibtex @research article { konuralpjournalmath418208, journal = {Konuralp Journal of Mathematics (KJM)}, issn = {}, eissn = {2147-625X}, address = {Mehmet Zeki SARIKAYA}, year = {2018}, volume = {6}, pages = {117 - 127}, doi = {}, title = {Injective and Relative Injective Zagreb Indıces of Graphs}, key = {cite}, author = {Alqesmah, Akram and Alwardi, Anwar and Rangarajan, R.} }
APA Alqesmah, A , Alwardi, A , Rangarajan, R . (2018). Injective and Relative Injective Zagreb Indıces of Graphs. Konuralp Journal of Mathematics (KJM), 6 (1), 117-127. Retrieved from http://dergipark.org.tr/konuralpjournalmath/issue/31478/418208
MLA Alqesmah, A , Alwardi, A , Rangarajan, R . "Injective and Relative Injective Zagreb Indıces of Graphs". Konuralp Journal of Mathematics (KJM) 6 (2018): 117-127 <http://dergipark.org.tr/konuralpjournalmath/issue/31478/418208>
Chicago Alqesmah, A , Alwardi, A , Rangarajan, R . "Injective and Relative Injective Zagreb Indıces of Graphs". Konuralp Journal of Mathematics (KJM) 6 (2018): 117-127
RIS TY - JOUR T1 - Injective and Relative Injective Zagreb Indıces of Graphs AU - Akram Alqesmah , Anwar Alwardi , R. Rangarajan Y1 - 2018 PY - 2018 N1 - DO - T2 - Konuralp Journal of Mathematics (KJM) JF - Journal JO - JOR SP - 117 EP - 127 VL - 6 IS - 1 SN - -2147-625X M3 - UR - Y2 - 2019 ER -
EndNote %0 Konuralp Journal of Mathematics (KJM) Injective and Relative Injective Zagreb Indıces of Graphs %A Akram Alqesmah , Anwar Alwardi , R. Rangarajan %T Injective and Relative Injective Zagreb Indıces of Graphs %D 2018 %J Konuralp Journal of Mathematics (KJM) %P -2147-625X %V 6 %N 1 %R %U
ISNAD Alqesmah, Akram , Alwardi, Anwar , Rangarajan, R. . "Injective and Relative Injective Zagreb Indıces of Graphs". Konuralp Journal of Mathematics (KJM) 6 / 1 (April 2018): 117-127.
AMA Alqesmah A , Alwardi A , Rangarajan R . Injective and Relative Injective Zagreb Indıces of Graphs. Konuralp J. Math.. 2018; 6(1): 117-127.
Vancouver Alqesmah A , Alwardi A , Rangarajan R . Injective and Relative Injective Zagreb Indıces of Graphs. Konuralp Journal of Mathematics (KJM). 2018; 6(1): 127-117.