Sharp lower bounds for the Zagreb indices of unicyclic graphs

The first Zagreb index $M_1$ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index $M_2$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper we present the lower bound on $M_1$ and $M_2$ among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which the bound is attained. Moreover, we obtain some relations between the Zagreb indices for unicyclic graphs.

Anahtar Kelimeler:

First Zagreb index, second Zagreb index, unicyclic graph, maximum degree, cycle length

Sharp lower bounds for the Zagreb indices of unicyclic graphs

Keywords:

First Zagreb index, second Zagreb index, unicyclic graph, maximum degree, cycle length,

PDF

___

Balaban AT, Motoc I, Bonchev D, Mekenyan O. Topological indices for structure–activity correlations. Topics Curr Chem 1983; 114: 21–55.
Belardo F, Marzi EML, Simi´c SK. Some results on the index of unicyclic graphs. Linear Algebra Appl 2006; 416: 1048–1059.
Caporossi G, Hansen P, Vukiˇcevi´c D. Comparing Zagreb indices of cyclic graphs. MATCH Commun Math Comput Chem 2010; 63: 441–451. [4] Das KC, Gutman I. Some properties of the second Zagreb index. MATCH Commun Math Comput Chem 2004; 52: 103–112.
Das KC, Gutman I, Zhou B. New upper bounds on Zagreb indices. J Math Chem 2009; 46: 514–521.
Deng H. A unified approach to the extremal Zagreb indices for trees, unicyclic graphs and bicyclic graphs. MATCH Commun Math Comput Chem 2007; 57: 597–616.
Gutman I, Das KC. The first Zagreb indices 30 years after. MATCH Commun Math Comput Chem 2004; 50: 83–92. [8] Gutman I, Ruˇsˇci´c B, Trinajsti´c N, Wilcox CF. Graph theory and molecular orbitals. XII. Acyclic polyenes. J Chem Phys 1975; 62: 3399–3405.
Gutman I, Trinajstic N. Graph theory and molecular orbitals. Total π -electron energy of alternant hydrocarbons. Chem Phys Lett 1972; 17: 535–538.
Hansen P, Vukiˇcevi´c D. Comparing the Zagreb indices. Croat Chem Acta 2007; 80: 165–168.
Horoldagva B, Das KC. Comparing variable Zagreb indices for unicyclic graphs. MATCH Commun Math Comput Chem 2009; 62: 725–730. [12]Horoldagva B, Lee S-G. Comparing Zagreb indices for connected graphs. Discrete Appl Math 2010; 158: 1073–1078. [13] Nikoli´c S, Kovaˇcevi´c G, Mili´cevi´c A, Trinajsti´c N. The Zagreb indices 30 years after. Croat Chem Acta 2003; 76: 113–124.
Sun L, Chen T. Comparing the Zagreb indices for graphs with small diﬀerence between the maximum and minimum degrees. Discrete Appl Math 2009; 157: 1650–1654.
Todeschini R, Consonni V. Handbook of Molecular Descriptors. Weinheim, Germany: Wiley–VCH, 2000.
Vukiˇcevi´c D, Graovac A. Comparing Zagreb M1 and M2 indices for acyclic molecules. MATCH Commun Math Comput Chem 2007; 57: 587–590.
Yan Z, Liu H, Liu H. Sharp bounds for the second Zagreb index of unicyclic graphs. J Math Chem 2007; 42: 565–574.
Zhang H, Zhang S. Unicyclic graphs with the first three smallest and largest first Zagreb index. MATCH Commun Math Comput Chem 2006; 55: 427–438.
Zhou B, Gutman I. Further properties of Zagreb indices. MATCH Commun Math Comput Chem 2005; 54: 233–239.