Abstract
Gutman defined the energy \( \mathcal E(G) \) of a simple graph \( G \) with vertex set \( V(G)=\{v_1,v_2,\ldots,v_n\} \) and edge
set \( E(G)=\{e_1,e_2,\ldots,e_m\} \) as the sum of the absolute values of eigenvalues of the adjacency matrix of \( G \), which
has been studied extensively in mathematical chemistry. In this note, we consider the relation between the energies of
\( G \) and the edge-deleted subgraphs of \( G \) and prove that for any positive integer \( k\leq m-1 \),
\[
\dbinom{m-1}{k}\mathcal E(G)\leq \sum_{M\in \Phi_k(G)}\mathcal E(G-M)\,,
\]
where \( \Phi_k(G)=\{M\subset E(G)||M|=k\} \). As corollary, we show that if \( m\geq 2 \), then
\[
\mathcal E(G)\leq \sqrt{2}m+\frac{4-2\sqrt{2}}{m-1}p(G;2)=2m-\frac{4-2\sqrt{2}}{m-1}\sum_{i=1}^n\dbinom{d_i}{2}\,,
\]
where \( p(G;2) \) is the number of \( 2 \)-matchings of \( G \) and \( d_i \) is the degree of \( v_i \) in \( G \).
