On the Stability of Independence Polynomials
On the Stability of Independence Polynomials
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DOI：
10.37236/7280
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发表时间：
2018-02
期刊：
影响因子：
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通讯作者：
Jason I. Brown;B. Cameron
中科院分区：
文献类型：
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作者：
Jason I. Brown;B. Cameron
图的独立多项式是每个大小的独立集个数的生成多项式，它的根称为独立根。我们研究了这种多项式的稳定性，即根位于左半平面的条件（独立多项式的所有真实的根都是负的，因此位于这个半平面）。我们证明了图的所有独立多项式的稳定性，独立数最多为3，但对于更大的独立数，我们证明了独立多项式可以有根任意远的权利。我们提供了家庭的图形，其独立多项式是稳定的，而不是，利用各种图形操作。
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions under which the roots lie in the left half-plane (all of the real roots of independence polynomial are negative and hence lie in this half-plane). We show stability for all independence polynomials of graphs with independence number at most three, but for larger independence number we show that the independence polynomials can have roots arbitrarily far to the right. We provide families of graphs whose independence polynomials are stable and ones that are not, utilizing various graph operations.