On the regularity of edge ideal of graphs

Let $G$ be a graph with $n$ vertices, $S=\mathbb{K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$ and $I(G)$ denote the edge ideal of $G$. For every collection $\mathcal{H}$ of connected graphs with $K_2\in \mathcal{H}$, we introduce the notions of $\text{ind}-matc{h}_{\mathcal{H}}(G)$ and $\min -matc{h}_{\mathcal{H}}(G)$. It will be proved that the inequalities $\ind-match_{{K_2, C_5}}(G)\leq{\rm reg}(S/I(G))\leq\min-match_{{K_2, C_5}}(G)$ are true. Moreover, we show that if $G$ is a Cohen–Macaulay graph with girth at least five, then ${\rm reg}(S/I(G))=\ind-match_{{K_2, C_5}}(G)$\mathrm{.}Furthermore,weprovethatif$G$isapaw--freeanddoublyCohen--Macaulaygraph,then${\rm reg}(S/I(G))=\ind-match_{{K_2, C_5}}(G)$ if and only if every connected component of $G$ is either a complete graph or a $5$-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality $\mathrm{r}\mathrm{e}\mathrm{g}(\mathbb{K}[\mathrm{\Delta }])=\mathrm{d}\mathrm{i}\mathrm{m}(\mathbb{K}[\mathrm{\Delta }])$ holds.