Graphs with maximal irregularity
Résumé
Albertson~\cite{Albertson} has defined the {\em irregularity} of a simple, undirected graph $G=(V,E)$ as
$irr(G) = \sum_{uv\in E}\left|d_G(u)-d_G(v)\right|$,
where $d_G(u)$ denotes the degree of a vertex $u \in V$.
For arbitrary graphs with $n$ vertices,
he has obtained an asymptotically tight upper bound on the irregularity
of $4 n^3 /27.$ Here, by exploiting a different approach than in~\cite{Albertson},
we show that for arbitrary graphs with $n$ vertices
the upper bound $\lfloor \frac{n}{3} \rfloor \lceil \frac{2 n}{3} \rceil \left( \lceil \frac{2 n}{3} \rceil -1\right)$
is sharp. We also determine graphs with maximal irregularity among $k-$cyclic graphs with
$n$ vertices.