Geodesic
An upper bound on the number of edges of a~graph which $k$-th power has connected complement
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VIII, Tome 450 (2016), pp. 151-174

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We call a graph $k$-wide, if for any division of its vertex set into two sets one can choose vertices of distance at least $k$ in these sets (i.e., the complement of $k$-th power of this graph is connected). We call a graph $k$-mono-wide, if for any division of its vertex set into two sets one can choose vertices of distance exactly $k$ in these sets. We prove that the complement of a $3$-wide graph on $n$ vertices has at least $3n-7$ edges and the complement of a $3$-mono-wide graph on $n$ vertices has at least $3n-8$ edges. We construct infinite series of graphs for which these bounds are attained. We also prove an asymptotically tight bound for the case $k\ge4$: the complement of a $k$-wide graph contains at least $(n-2k)(2k-4[\log_2k]-1)$ edges. 
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     author = {V. S. Samoilov},
     title = {An upper bound on the number of edges of a~graph which $k$-th power has connected complement},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {151--174},
     publisher = {mathdoc},
     volume = {450},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_450_a7/}
}
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V. S. Samoilov. An upper bound on the number of edges of a~graph which $k$-th power has connected complement. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VIII, Tome 450 (2016), pp. 151-174. http://geodesic.mathdoc.fr/item/ZNSL_2016_450_a7/