New lower bounds for the first variable Zagreb index

The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^\alpha$, and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the first Zagreb, the inverse and the forgotten indices.

[1]  H. Narumi,et al.  Simple Topological Index : A Newly Devised Index Characterizing The Topological Nature of Structural Isomers of Saturated Hydrocarbons , 1984 .

[2]  M. Drmota.  Random Trees: An Interplay between Combinatorics and Probability , 2009 .

[3]  Bolian Liu,et al.  Some properties of the first general Zagreb index , 2010, Australas. J Comb..

[4]  D. Vukicevic,et al.  Bond Additive Modeling 2. Mathematical Properties of Max-min Rodeg Index , 2010 .

[5]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[6]  José M. Rodríguez,et al.  New lower bounds for the second variable Zagreb index , 2018, J. Comb. Optim..

[7]  M. Randic.  On computation of optimal parameters for multivariate analysis of structure‐property relationship , 1991 .

[8]  T. F. Móri.  On random trees , 2002 .

[9]  M. Randic.  Characterization of molecular branching , 1975 .

[10]  D. Cvetkovic,et al.  Graph theory and molecular orbitals , 1974 .

[11]  N. Trinajstic,et al.  The Zagreb Indices 30 Years After , 2003 .

[12]  D. Vukicevic,et al.  Bond Additive Modeling 1. Adriatic Indices , 2010 .

[13]  M. Randic.  Novel graph theoretical approach to heteroatoms in quantitative structure—activity relationships , 1991 .

[14]  Shenggui Zhang,et al.  Bicyclic graphs with the first three smallest and largest values of the first general Zagreb index , 2006 .

[15]  Anthony J. Guttmann,et al.  On the Number of Benzenoid Hydrocarbons , 2002, J. Chem. Inf. Comput. Sci..

[16]  I. Gutman,et al.  Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices , 2013 .

[17]  Milan Randic,et al.  Variable Connectivity Index for Cycle-Containing Structures , 2001, J. Chem. Inf. Comput. Sci..

[18]  S. Nikolic,et al.  On Variable Zagreb Indices , 2004 .

[19]  P. Alam.  ‘A’ , 2021, Composites Engineering: An A–Z Guide.

[20]  Jos'e M. Rodr'iguez,et al.  New lower bounds for the Geometric-Arithmetic index , 2017, 1703.05524.

[21]  I. Gutman.  Degree-Based Topological Indices , 2013 .

[22]  I. Gutman,et al.  Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons , 1972 .

[23]  MODIFIED NARUMI-KATAYAMA INDEX , 2012 .