New Sharp lower bounds for the first Zagreb index
Authors: Mansour T., Rostami M.A., Suresh E., Xavier G.B.A.
Keywords: first Zagreb index; second Zagreb index; inverse degree
Abstract:
The first Zagreb index M1 (G) defined as the sum of squares of the degrees of the vertices. In this paper we compare and analyze numerous lower bounds for the first Zagreb index involving the number of vertices, the number of edges and the maximum and minimum vertex degree. In addition, we propose new lower bound and correct the equality case in [E.I. Milovanović and I.Ž. Milovanović, Sharp Bounds for the first Zagreb index and first Zagreb coindex, Miskolc Mathematical notes, 16 (2015) 1017-1024].
References:
[1] Bianchi, M., Cornaro, A., Palacios, J.L., Torriero, A. (2015) New bounds of degree-based topological indices for some classes of c-cyclic graphs. Discrete App. Math., 184, 62-75
[2] Das, K.Ch. (2003) Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac Journal of Mathematics, br. 25, str. 31-49
[3] Das, K.Ch., Xu, K., Nam, J. (2015) Zagreb indices of graphs. Frontiers of Mathematics in China, 10(3): 567-582
[4] de Caen, D. (1998) An upper bound on the sum of squares of degrees in a graph. Discrete Mathematics, 185(1-3): 245-248
[5] Edwards, C.S. (1977) The Largest Vertex Degree Sum for a Triangle in a Graph. Bulletin of the London Mathematical Society, 9(2): 203-208
[6] Fajtlowicz, S. (1987) On conjectures of graffiti. Congr. Numer, II, 60, 189-197
[7] Gutman, I., Trinajstić, N. (1972) Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4): 535-538
[8] Ilic, A., Stevanovic, D.P. (2009) On Comparing Zagreb Indices. MATCH commun. math. comput. chem., =vol. 62, br. 3, str. 681-687
[9] Milovanovic, E., Milovanovic, I. (2015) Sharp bounds for the first Zagreb index and first Zagreb coindex. Miskolc Mathematical Notes, 16(2): 1017-1024
[10] Rostami, M. A., Bücker, H. M., Azadi, A. (2014) Illustrating a Graph Coloring Algorithm Based on the Principle of Inclusion and Exclusion Using GraphTea. Lecture Notes in Computer Science, str. 514-517
[11] Xu, K., Das, K.C. (2015) Some extremal graphs with respect to inverse degree. Discrete App. Math
[12] Zhou, B., Trinajstić, N. (2010) On general sum-connectivity index. Journal of Mathematical Chemistry, 47(1): 210-218