The average eccentricity of a graph  G , denoted by  ecc ( G ), is the mean value of eccentricities of all vertices of  G . Let  D   n, i   be the  n   -vertex tree obtained from a path   P n−1 =v 1 v 2 ⋯v n−1        P   n  −  1    =   v  1    v  2   ⋯   v   n  −  1           by attaching a pendent vertex to  v i  . In [13], it was shown that the maximum value for the sum of domination number and average eccentricity among  n -vertex (connected) graphs is attained by  D   n , 3  when             n  ≡  0    (    mod      3  )  ,         and attained by the path  P n     when             n  ¬  ≡  0    (    mod      3  )        . In this paper, we will further determine the second maximum value for the sum of domination number and average eccentricity among  n -vertex (connected) graphs. It is interesting that the graphs attaining that second maximum value have three cases, which is  D   n , 6  when             n  ≡  0    (    mod      3  )  ,         D   n , 3  when             n  ≡  1    (    mod      3  )  ,         and  T n     when             n  ≡  2    (    mod      3  )  ,         where  T n   is the  n   -vertex tree obtained from a path   P n−2 =v 1 v 2 ⋯v n−2        P   n  −  2    =   v  1    v  2   ⋯   v   n  −  2           by attaching a pendent vertex to  v  3 , and a pendent vertex to   v n−4       v   n  −  4         .
['Zhibin Du']
Further results regarding the sum of domination number and average eccentricity