On the Wiener Index and Total Eccentricity Index

In this paper, some bounds for the total eccentricity index and for the Wiener is proposed. It is proved that for the graphs of order at least 5, the Wiener index is greater than the total eccentricity index and the equality holds if and only if the graph is a cycle of order 4 or 3 or is a path of order 4. Moreover some bound is given on the difference the Wiener index and total eccentricity index. It is shown that among all trees of a same order, the difference of the Wiener index and the total eccentricity index is minimized when the graph is a caterpillar.