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In trees with n vertices, the Wiener index of tree is minimized by stars and maximized by paths, both uniquely. In this paper, we give an inequality similar in the case of planar maps.
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... 1. Let be a map with |V|=5, we have: deg( ) = 4, deg( ) = 2, W( , ) = 4, W( , ) = 6, and are complete vertices (see Fig 2). where C − is the map obtained by deleting the edge of the map C. We notice that delete a multiple edge does not affect the Wiener index; through this paper, we consider only the simple maps (without loops and without multiple edges). Lemma 2. Let be a simple map with n vertices and let υ be a complete vertex of C, then Lemma 3. Let be a simple map with n vertices (n ≥ 2) and let υ a vertex not complete of a map C, then 1. Let C be a simple map and let υ a vertex of map , then W( υ , ) ≥ n-1. 2. Let be a simple map with n vertices, e be an edge of and let be the map obtained by deleting the edge e such that the map remains connected, then W ( ) ≥ W( ). Theorem 1. Let be a simple map with n vertices, ...
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