Geodesic
Minimum Eccentric Connectivity Index for Graphs With Fixed Order and Fixed Number of Pendant
Yugoslav journal of operations research, Tome 29 (2019) no. 2, p. 193
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The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product $d_G(v) e_G(v)$, where $d_G(v)$ is the degree of $v$ in $G$ and $e_G(v)$ is the maximum distance between $v$ and any other vertex of $G$. We characterize, with a new elegant proof, those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order $n$. Then, given two integers $n$ and $p$ with $p \le n-1$, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order $n$ with $p$ pendant vertices.
Classification : 05C35, 05C40
Keywords: Extremal Graph Theory, Eccentric Connectivity Index, Pendant Vertices 
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     author = {Gauvain Devillez and Alain Hertz and Hadrien M\'elot and Pierre Hauweele},
     title = {Minimum {Eccentric} {Connectivity} {Index} for {Graphs} {With} {Fixed} {Order} and {Fixed} {Number} of {Pendant}},
     journal = {Yugoslav journal of operations research},
     pages = {193 },
     year = {2019},
     volume = {29},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/YJOR_2019_29_2_a2/}
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Gauvain Devillez; Alain Hertz; Hadrien Mélot; Pierre Hauweele. Minimum Eccentric Connectivity Index for Graphs With Fixed Order and Fixed Number of Pendant. Yugoslav journal of operations research, Tome 29 (2019) no. 2, p. 193 . http://geodesic.mathdoc.fr/item/YJOR_2019_29_2_a2/