Further results regarding the sum of domination number and average eccentricity

The average eccentricity of a graph G, denoted by ecc(G), is the mean value of eccentricities of all vertices of G. Let Dn, i be the n-vertex tree obtained from a path P n - 1 = v 1 v 2 ? v n - 1 by attaching a pendent vertex to vi. 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 Dn, 3 when n ? 0 ( mod 3 ) , and attained by the path Pn 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 Dn, 6 when n ? 0 ( mod 3 ) , Dn, 3 when n ? 1 ( mod 3 ) , and Tn when n ? 2 ( mod 3 ) , where Tn is the n-vertex tree obtained from a path P n - 2 = v 1 v 2 ? v n - 2 by attaching a pendent vertex to v3, and a pendent vertex to v n - 4 .

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