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In this paper we prove the conjectured upper bound for CastelnuovoâMumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques
of G, denoted by c(G). Moreover, we classify all chordal graphs G for which L(G) = c(G), where L(G) is the sum of the lengths of longest induced paths of connected components of G. We call such graphs strong interval graphs. We show that the regularity of a strong
interval graph G coincides with L(G) as well as c(G).
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