Bounds for Energy of a Graph with Self-Loops

Let G_S be a graph with σ self-loops obtained from the graph G by attaching a self-loop to each vertex in the subset S⊆ V(G) and |S|=σ. The energy of a graph G_S is defined as (Formula presented), where λ₁,λ₂,…,λ_n are the eigenvalues of the adjacency matrix of a graph G_S. In this paper, we established a lower bound for the spectral radius of matrix (Formula presented) in terms of number of vertices (n), number of edges (m), and minimum vertex degree (δ). Additionally, we derive several upper and lower bounds for the energy of a graph with self-loops in terms of n, m, number of self-loops (σ), maximum vertex degree (Δ), and δ.