The minimum exponential atom-bond connectivity energy of trees

Abstract

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

Keywords

• tree
• exponential atom-bond connectivity index
• exponential atom-bond connectivity energy
• 05c50
• 05c09
• 05c92