Geodesic
Bipartite Graphs with the Maximal Value of the Second Zagreb Index
Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 1 Cet article a éte moissonné depuis la source Bulletin of the Malaysian Mathematical Society website

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The second Zagreb index of a graph $G$ is an adjacency-based topological index, which is defined as $\sum_{uv \in E(G)}(d(u)d(v))$, where $uv$ is an edge of $G$, $d(u)$ is the degree of vertex $u$ in $G$. In this paper, we consider the second Zagreb index for bipartite graphs. Firstly, we present a new definition of ordered bipartite graphs, and then give a necessary condition for a bipartite graph to attain the maximal value of the second Zagreb index. We also present an algorithm for transforming a bipartite graph to an ordered bipartite graph, which can be done in $O(n_2+n_1^2)$ time for a bipartite graph $B$ with a partition $|X|=n_1$ and $|Y|=n_2$.
Classification : 05C35 
@article{BMMS_2013_36_1_a0,
     author = {Rongling Lang and Xiaole Deng and Hui Lu},
     title = {Bipartite {Graphs} with the {Maximal} {Value} of the {Second} {Zagreb} {Index}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2013},
     volume = {36},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2013_36_1_a0/}
}
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Rongling Lang; Xiaole Deng; Hui Lu. Bipartite Graphs with the Maximal Value of the Second Zagreb Index. Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 1. http://geodesic.mathdoc.fr/item/BMMS_2013_36_1_a0/