The average Steiner \(( 3 , 2 )\)-eccentricity of trees (Q6559393)

Let \(G\) be a graph and let \(S\subseteq V(G)\) be a set with three vertices. The Steiner distance \(d_G(S)\) is the minimum number of edges on a subtree of \(G\) that contains all three vertices of \(S\). The Steiner 3-eccentricity of a vertex \(v\in V(G)\) is \[\varepsilon_3(v;G)=\max\{d_G(S):v\in S\subseteq V(G), |S|=3\}.\] The Steiner \((3,2)\)-eccentricity is a generalization from one vertex \(v\) in \(\varepsilon_3(v;G)\) to two vertex subset \(S^\prime\) defined by \[\varepsilon_{3,2}(S^\prime;G)=\max\{d_G(S):S^\prime\subset S\subseteq V(G), |S^\prime|=2,|S|=3\}.\] Now, the average Steiner \((3,2)\)-eccentricity is the arithmetical mean value of Steiner \((3,2)\)-eccentricities over all vertices of \(G\): \[\overline{\varepsilon}_{3,2}(G)=\binom{|V(G)|}{2}^{-1}\sum_{S^\prime\subseteq V(G)}\varepsilon_{3,2}(S^\prime;G).\]\N\NThe main results of this contribution are several (upper and lower) bounds for \(\overline{\varepsilon}_{3,2}(T)\) of a tree \(T\) with respect to several fix parameters (order, number of leaves and maximum degree). Along with this, trees are described which achieve these bounds. The main tool in the proofs is a transformation and its inverse defined on trees.