Variations of the eccentricity and their properties in trees

• Introduce several new distance-based global and local functions based the smallest distance which are related to eccentricity, center, sum of eccentricities in graphs trees. The values acompared with some sharp bounds established. difference between eccentricity uniformity behaves a very similar way as itself. Motivated from study trees, we introduce on vertex leaf (called “uniformity” at that vertex). Some natural extremal problems trees considered. Then middle parts tree is discussed compared well-known center tree. also Last but not least, show uniformity, when considered function,