Relations between ordinary energy and energy of a self-loop graph
B.R. Rakshith,
Kinkar Chandra Das,
B.J. Manjunatha,
Yilun Shang
Affiliations
B.R. Rakshith
Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576 104, India
Kinkar Chandra Das
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea; Corresponding authors.
B.J. Manjunatha
Department of Mathematics, Sri Jayachamarajendra College of Engineering, JSS Science and Technology University, Mysuru–570 006, India; Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru-570 002, Affiliated to Visvesvaraya Technological University, Belagavi-590 018, India
Yilun Shang
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK; Corresponding authors.
Let G be a graph on n vertices with vertex set V(G) and let S⊆V(G) with |S|=α. Denote by GS, 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 GS (E(GS)) in terms of ordinary energy (E(G)), order (n) and number of self-loops (α). Recently, it is proved that for a bipartite graph GS, E(GS)≥E(G). Here we show that this inequality is strict for an unbalanced bipartite graph GS with 0<α<n. In other words, we show that there exits no unbalanced bipartite graph GS with 0<α<n and E(GS)=E(G).
