Clique polynomials have a unique root of smallest modulus

Entropy and the clique polynomial

“…, v j commute with a; cf. [GS,§4]. In this case, we could reorder the letters as av i · · · v j to obtain a g < g with…”

On the roots of independence polynomials of almost all very well-covered graphs

“…Denoting by min , max the smallest and the largest real root of I (G; x), respectively, we get that min max < 0, since all the coefficients of I (G; x) are positive. The following proposition summarizes results from [13,16,2,18]. It is also shown in [2] that for any well-covered graph G there is a well-covered graph H with (G) = (H ) such that G is an induced subgraph of H and I (H ; x) has all its roots simple and real.…”

“…Moreover, it is easy to check that the complete n-partite graph G = K , ,..., is well-covered, (G) = , and its independence polynomial I (G; x) = n(1 + x) − (n − 1) has only one real root, whenever is odd, and exactly two real roots, for any even 2. The roots of the independence polynomial of (well-covered) graphs are investigated in a number of papers, as [2][3][4][5]13,16,18]. Denoting by min , max the smallest and the largest real root of I (G; x), respectively, we get that min max < 0, since all the coefficients of I (G; x) are positive.…”

“…where s k is the number of stable sets of size k in G. In [16], D(G; x) is defined as the clique polynomial of G. Clique polynomials are related to trace monoids. In fact, 1/D(G; x) is the generating function of the sequence of the number of traces of different sizes in the trace monoid defined by G, see [15,16].…”

“…In fact, 1/D(G; x) is the generating function of the sequence of the number of traces of different sizes in the trace monoid defined by G, see [15,16].…”