Characterization of Extremal Unicyclic Graphs with Fixed Leaves Using the Lanzhou Index

Abstract

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical temperature, surface tension, boiling point, octanol-water partition coefficient, density and flash points. For a (molecular) graph Γ, the Lanzhou index (Lz index) is obtained by the sum of deg(v)2de¯g(v) over all the vertices, where deg(v) and de¯g(v) are degrees of the vertex v in Γ and its complement Γ¯ respectively. Let Vαβ be a class of unicyclic graphs (same order and size) such that each graph of this class has order α and β leaves (vertices of degree one). In this note, we compute the lower and upper bounds of Lz index for each unicyclic graph in the class of graphs Vαβ. Moreover, we characterize the extremal graphs with respect to Lz index in the same class of graphs.