The minimum exponential atom-bond connectivity energy of trees

Abstract

Abstract Let G = ( V ( G ) , E ( G ) ) G=\left(V\left(G),E\left(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{\rm{ABC}}}}\left(G) of G G is an n × n n\times n matrix whose ( i , j ) \left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{\sqrt{\tfrac{d\left({v}_{i})+d\left({v}_{j})-2}{d\left({v}_{i})d\left({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}\in E\left(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{\rm{ABC}}}}\left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.