Relations between ordinary energy and energy of a self-loop graph

B R Rakshith; Kinkar Chandra Das; B J Manjunatha; Yilun Shang

doi:10.1016/j.heliyon.2024.e27756

Relations between ordinary energy and energy of a self-loop graph

Heliyon. 2024 Mar 8;10(6):e27756. doi: 10.1016/j.heliyon.2024.e27756. eCollection 2024 Mar 30.

Authors

B R Rakshith¹, Kinkar Chandra Das², B J Manjunatha^{3

4}, Yilun Shang⁵

Affiliations

¹ Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576 104, India.
² Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea.
³ Department of Mathematics, Sri Jayachamarajendra College of Engineering, JSS Science and Technology University, Mysuru-570 006, India.
⁴ Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru-570 002, Affiliated to Visvesvaraya Technological University, Belagavi-590 018, India.
⁵ Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK.

Abstract

Let G be a graph on n vertices with vertex set $V (G)$ and let $S \subseteq V (G)$ with $| S | = α$ . Denote by $G_{S}$ , the graph obtained from G by adding a self-loop at each of the vertices in S. In this note, we first give an upper bound and a lower bound for the energy of $G_{S}$ ( $E (G_{S})$ ) in terms of ordinary energy ( $E (G)$ ), order (n) and number of self-loops (α). Recently, it is proved that for a bipartite graph $G_{S}$ , $E (G_{S}) \geq E (G)$ . Here we show that this inequality is strict for an unbalanced bipartite graph $G_{S}$ with $0 < α < n$ . In other words, we show that there exits no unbalanced bipartite graph $G_{S}$ with $0 < α < n$ and $E (G_{S}) = E (G)$ .

Keywords: 05C50; Energy of a graph; Graphs with self-loops; Singular values.