On linear combinations between Zagreb indices/coindices of a line graph

Document Type : Special Issue: Advances in Combinatorics and Graph Theory: From Algebraic Structures to Real-World Applications

Authors

Faculty of Electronic Engineering, University of Nis, Nis, Serbia

Abstract

Let $G=(V,E)$, $V=\left\{ v_{1},v_{2},\ldots ,v_{n}\right\}$, be a simple graph of order $n$ and size $m$.
Denote by $\Delta = d_1\ge d_2 \ge \cdots \ge d_n= \delta$, $d_i=d(v_i)$, and $\Delta_e=d(e_1)\ge d(e_2)\ge \cdots \ge d(e_m)=\delta_e$, sequences of vertex and edge degrees, respectively. The first reformulated Zagreb index (coindex) is defined as $\displaystyle EM_1(G)=\sum_{i=1}^m d(e_i)^2 = \sum_{e_i\sim e_j}(d(e_i)+d(e_j))$ $\Big(\displaystyle \overline{EM}_1(G) = \sum_{e_i\nsim e_j}(d(e_i)+d(e_j))\Big)$.
We consider relationship between reformulated Zagreb indices/coindices and determine their bounds in terms of some basic graph parameters.

Graphical Abstract

On linear combinations between Zagreb indices/coindices of a line graph

Keywords

Main Subjects


[1] A. Ali, I. Gutman, E. Milovanovi´c, I. Milovanovi´c, Sum of powers of the degrees of graphs: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018) 5–84.
[2] A. R. Ashrafi, T. Doˇsli´c, A. Hamzeh, The Zagreb coindices of graph operations, Discr. Appl. Math. 158 (2010) 1571–1578.
[3] B. Bollob´as, P. Erd¨ os, Graphs of extremal weights, Ars Comb. 50 (1998) 225–233.
[4] B. Borovi´canin, K. C. Das, B. Furtula, I. Gutman, Bounds for the Zagreb indices,MATCH Commun. Math. Comput. Chem. 78(1) (2017) 17–100.
[5] B. Borovi´canin, K. C. Das, B. Furtula, I. Gutman, Zagreb indices: Bounds and extremal graphs, in:
I. Gutman, B. Furtula, K. C. Das, E. Milovanovi´c, I. Milovanovi´c (Eds.), Bounds in Chemical Graph
Theory – Basics, Univ. Kragujevac, Kragujevac, 2017, 67–153.
[6] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discr. Math. 285 (2004)
57–66.
[7] K. C. Das, Sharp bounds for the sum of the squares of degrees of a graph, Kragujevac J. Math. 25 (2003) 31–49.
[8] N. De, Some bounds of reformulated Zagreb indices, Appl. Math. Sci. 6(101) (2012) 5005–5012.
[9] T. Doˇsli´c, Vertex–weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1
(2008) 66–80.
[10] M. Gordon, G. J. Scantelbury, Non-random polycondensation: statistical theory of substitution
effect, Trans. Farady Soc. 60 (1964) 604–621.
[11] I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals. Total π-electron energy of alternate hydrocarbons, Chem. Phys. Lett. 17(4) (1972) 535–538.
[12] I. Gutman, B. Ruˇsˇci´c, N. Trinajsti´c, C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 3399–3405.
[13] I. Gutman, K. Ch. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput.
Chem. 50 (2004) 83–92.
[14] I. Gutman, E. Milovanovi´c, I. Milovanovi´c, Beyond the Zagreb indices, AKCE Int. J. Graphs Comb. 17(1) (2020) 74–85.
[15] A. Ili´c, B. Zhou, On reformulated Zagreb indices, Discr. Appl. Math. 160 (2012) 204–209.
[16] B. Liu, I. Gutman, Upper bounds for the Zagreb indices of connected graphs, MATCH Commun. Math. Comput. Chem. 55 (2006) 439–446.
[17] T. Mansour, M. A. Rostami, E. Suresh, G. B. A. Xavier, New sharp lower bounds for the first Zagreb index, Sci. Publ. State Univ. Novi Pazar, Ser: A, Appl. Math. Inform. Mech. 8 (1) (2016) 11–19.
[18] A. Miliˇcevi´c, S. Nikoli´c, N. Trinajsti´c, On reformulated Zagreb indices, Mol. Diversity 8 (2004)
393–399.
[19] E. I. Milovanovic´, I. Zˇ . Milovanovic´, E. C´ . Dolic´anin, E. Glogic´, A note on the first reformulated Zagreb index, Appl. Math. Comput. 273 (2016) 16–20.
[20] E. I. MIlovanovic´, I. Zˇ . Milovanovic´, Sharp bounds for the first Zagreb index and first Zagreb
coindex, Miskolc Math. Notes 16 (2015) 1017–1024.
[21] D. S. Mitrinovi´c, P. M. Vasi´c, Analytic inequalities, Springer Verlag, Berlin–Heidelberg–New York, 1970.
[22] S. Nikoli´c, G. Kovaˇcevi´c, A. Miliˇcevi´c, N. Trinajsti´c, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124.
[23] K. Pattabiraman, A. Santhakumar, Bounds on first reformulated Zagreb index of graphs, Caspian J. Math. Sci. 7(1) (2018) 25–35.
[24] J. R. Platt, Influence of neighbor bonds on additive bond properties in paraffins, J. Chem. Phys. 15 (1947) 419–420.
[25] B. Zhou, N. Trinajsti´c, Some properties of the reformulated Zagreb indices, J. Math. Chem. 48
(2010) 714–719.
Volume 8, Issue 2
July 2023
Pages 63-72
  • Receive Date: 27 April 2023
  • Revise Date: 03 May 2023
  • Accept Date: 15 May 2023
  • Publish Date: 01 June 2023