Let G be a finite simple graph on n vertices, that contains no isolated vertices, and let \(I(G) \subseteq S = K[x_1, \dots , x_n]\) be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of S/I(G). We show that if \({{\,\mathrm{pd}\,}}(S/I(G))\) attains its minimum possible value \(2\sqrt{n}-2\) then, with only one exception, \({{\,\mathrm{reg}\,}}(S/I(G)) = 1\). We also provide a full description of the spectrum of \({{\,\mathrm{pd}\,}}(S/I(G))\) when \({{\,\mathrm{reg}\,}}(S/I(G))\) attains its minimum possible value 1.

令G为n个顶点上的有限简单图，其中不包含孤立的顶点，令\（I（G）\ subseteq S = K [x_1，\ dots，x_n] \）为边缘理想。在本文中，我们研究了测量投影尺寸和S / I（G）正则性的一对整数。我们证明，如果\（{{\，\ mathrm {pd} \，}}（S / I（G））\）达到其最小可能值\（2 \ sqrt {n} -2 \），则仅一个例外，\（{{\，\ mathrm {reg} \，}}（S / I（G））= 1 \）。当\（{{\\ mathrm {reg} \，}时，我们也提供\（{{\，\ mathrm {pd} \，}}（S / I（G））\）的频谱的完整描述。}（S / I（G））\） 达到其最小可能值1。