Journal of Prime Research in Mathematics

# Bounds of F-index for unicyclic graphs with fixed pendent vertices

M. Javaid
Department of Mathematics, School of Science, University of Management and Technology, Lahore, Pakistan.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan.
M. Hussain
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Malaysia.
W.C. Teh
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Malaysia.

$$^{1}$$Corresponding Author: javaidmath@gmail.com

### Abstract

Furtula and Gutman [J. Math. Chem., 53 (4) (2015), 1184- 1190] reinvestigated the $$F$$-index as a sum of cubes of the degrees of all the vertices in a chemical graph and proved its various properties. A connected graph with equal order and size is called unicyclic graph, where order is number of vertices and size is number of edges. In this paper, we characterize the extremal graphs in a family of graphs called by unicyclic graphs with fixed number of pendent vertices. We also investigate the bound on $$F$$-index in the same family of graphs i.e $$4(2n + 3α) ≤ F(G) ≤ 8n + α(α + 2)(α + 3)$$ for each $$G ∈ \mathcal{U}_{n}^{ α}$$, where $$\mathcal{U}_{n}^{ α}$$ is a class of all the unicyclic graphs such that the order of each graph is $$n$$ with $$α$$ pendent vertices.

#### Keywords:

Forgotten index, Unicyclic graphs, Extremal graphs.