On the Harary Estrada index of graphs

Abstract

Let GG be a connected graph with nn vertices v1,…,vn{v}_{1},\ldots ,{v}_{n}. The Harary matrix of GG, denoted by H(G)H\left(G), is an n×nn\times n matrix with a zero main diagonal, where the (i,j)\left(i,j)-entry is 1d(vi,vj)\frac{1}{d\left({v}_{i},{v}_{j})} for i≠ji\ne j, and d(vi,vj)d\left({v}_{i},{v}_{j}) represents the distance between vi{v}_{i} and vj{v}_{j}. Let ρ1,…,ρn{\rho }_{1},\ldots ,{\rho }_{n} be the eigenvalues of H(G)H\left(G). The Harary Estrada index of GG is defined as HEE(G)=∑i=1neρi{\rm{HEE}}\left(G)=\mathop{\sum }\limits_{i=1}^{n}{e}^{{\rho }_{i}}. In this article we study Harary Estrada index and find some new sharp bounds for the Harary Estrada index of graphs. Our results generalize and improve the previous bounds on the Harary Estrada index of graphs. Finally, we obtain some Nordhaus-Gaddum type inequalities for the Harary Estrada index of graphs.

Keywords

• harary matrix
• harary estrada index
• 05c09
• 05c50
• 05c92
• 15a18