Some notes on the energy of graphs with loops

Abstract

Let G be a graph of order n and (Formula presented.). The loop graph (Formula presented.) is a graph obtained from G by attaching a loop at each vertex in S. The energy of a loop graph is defined as (Formula presented.). It was conjectured that for every graph of order at least 2, there exists a subset (Formula presented.) such that (Formula presented.). In this paper, we prove this conjecture and obtain a stronger result by showing that for every integer r, 0<r<n, there exists (Formula presented.) such that (Formula presented.) and (Formula presented.). Also, we show that if G is a bipartite graph of odd order and (Formula presented.), then (Formula presented.). It is shown that for every graph G with nullity r, there exists (Formula presented.) such that (Formula presented.) and the adjacency matrix of (Formula presented.) is non-singular. © 2025 Informa UK Limited, trading as Taylor & Francis Group.