# Maximizing distance between center, centroid and subtree core of trees

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## Abstract

For \(n\ge 5\) and \(2\le g\le n-3,\) consider the tree \(P_{n-g,g}\) on *n* vertices which is obtained by adding *g* pendant vertices to one end vertex of the path \(P_{n-g}\). We call the trees \(P_{n-g,g}\) as path-star trees. The *subtree core* of a tree *T* is the set of all vertices *v* of *T* for which the number of subtrees of *T* containing *v* is maximum. We prove that over all trees on \(n\ge 5\) vertices, the distance between the center (respectively, centroid) and the subtree core is maximized by some path-star trees. We also prove that the tree \(P_{n-g_0,g_0}\) maximizes both the distances among all path-star trees on *n* vertices, where \(g_0\) is the smallest positive integer satisfying \(2^{g_0}+g_0>n-1\).

## Keywords

Tree center centroid subtree core distance## Mathematics Subject Classification

05C05 05C12## References

- 1.Harary F, Graph Theory (1969) (Addison-Wesley Publishing Co.)Google Scholar
- 2.Mitchell S L, Another characterization of the centroid of a tree,
*Discrete Math.***24**(1978) 277–280MathSciNetCrossRefGoogle Scholar - 3.Patra K L, Maximizing the distance between center, centroid and characteristic set of a tree,
*Linear Multilinear Algebra***55**(2007) 381–397MathSciNetCrossRefGoogle Scholar - 4.Székely L A and Wang H, On subtrees of trees,
*Adv. Appl. Math.***34**(2005) 138–155MathSciNetCrossRefGoogle Scholar - 5.Zelinka B, Median and peripherians of trees,
*Arch. Math.***4**(1968) 87–95MathSciNetzbMATHGoogle Scholar

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© Indian Academy of Sciences 2018