Graphs with smallest resolvent Estrada indices

Abstract

The graphs and trees with smallest resolvent Estrada indices (EEr ) are characterized. The connected graph of order n with smallest EEr -value is the n-vertex path. The second–smallest such graph is the (n − 1)-vertex path with a pendent vertex attached at position 2. The tree with third–smallest EEr is the (n − 1)-vertex path with a pendent vertex attached at position 3, conjectured to be also the connected graph with third–smallest EEr . Based on a computer–aided search, we established the structure of a few more trees with smallest EEr .