A note on (local) energy of a graph

Let G be a simple graph with vertex set V(G)(|V(G)|=n) and let S⊆V(G). We denote by d_i, the degree of the vertex v_i. The graph G^S is obtained from G by removing all the vertices belonging to S (If S={v_j}, then G^S is denoted by G^(j)). The energy of G is the sum of all absolute values of the eigenvalues of the adjacency matrix A(G) and is denoted by E(G). Recently, Espinal and Rada (MATCH Commun Math Comput Chem 92(1):89–103, 2024) introduced the concept of local energy of a graph e(G). It is defined as e(G)=∑_j=1ⁿE_G(v_j), where E_G(v_j)=E(G)-E(G^(j)) is called the local energy of a graph G at vertex v_j. In this paper, we prove that if v₁∈S and S is a vertex independent set of size k such that every vertex in S share the same open neighborhood set N_G(v₁), then E(G)-E(G^S)≤2kd₁. We also characterize graphs that satisfy the equality case. If S={v₁}, we get E(G)-E(G⁽¹⁾)≤2d₁ Espinal and Rada (MATCH Commun Math Comput Chem 92(1):89–103, 2024). One of the open problems in the study of local energy of a graph is to characterize graphs with e(G)=2E(G). Motivated by this problem, we present an infinite class of graphs for which e(G)<2E(G). As a result, we show that for a complete multipartite graph G, e(G)=2E(G) if and only if G≅K₂. We also prove that the local energy of a complete multipartite graph G is constant at each vertex of the graph if and only if G is regular. Finally, we give an upper bound on e(G) in terms of n and chromatic number k.