A note on (local) energy of a graph

Let G be a simple graph with vertex set \(V(G)\,(|V(G)|=n)\) and let \(S\subseteq V(G)\). 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 \({\mathcal {E}}(G)\). 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 \(e(G)=\sum ^n_{j=1}\,\mathcal {E}_{G}(v _j)\), where \(\mathcal {E}_{G}(v_j)=\mathcal {E}(G)-\mathcal {E}(G^{(j)})\) is called the local energy of a graph G at vertex \(v_j\). In this paper, we prove that if \(v_{1}\in S\) and S is a vertex independent set of size k such that every vertex in S share the same open neighborhood set \(N_{G}(v_{1})\), then \(\mathcal {E}(G)-\mathcal {E}(G^{S})\le 2\,\sqrt{k\,d_{1}}\). We also characterize graphs that satisfy the equality case. If \(S=\{v_{1}\}\), we get \(\mathcal {E}(G)-\mathcal {E}(G^{(1)})\le 2\,\sqrt{d_{1}}\) 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 \(e(G)=2\mathcal {E}(G)\). Motivated by this problem, we present an infinite class of graphs for which \(e(G)<2\mathcal {E}(G)\). As a result, we show that for a complete multipartite graph G, \(e(G)=2\mathcal {E}(G)\) if and only if \(G\cong K_{2}\). 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.