The independence polynomial i(G, x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of such polynomials. * Communicating author. Jason.Brown@dal.ca arXiv:1709.08236v1 [math.CO]
For a graph G, the mean subtree order of G is the average order of a subtree of G. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph G and every pair of distinct vertices u and v of G, the addition of the edge between u and v increases the mean subtree order. In fact, we show that the addition of a single edge between a pair of nonadjacent vertices in a graph of order n can decrease the mean subtree order by as much as n∕3 asymptotically. We propose the weaker conjecture that for every connected graph G which is not complete, there exists a pair of nonadjacent vertices u and v, such that the addition of the edge between u and v increases the mean subtree order. We prove this conjecture in the special case that G is a tree.
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called 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.where i k is the number of independent sets of size k in G. We call the roots of i(G, x) the independence roots of G.Research on the independence polynomial and in particular, the independence roots, has been very active (see, for example, [2,3,4,5,8,14] and [13] for an excellent survey) since it was first defined by Gutman and Harary in 1983 [11] (including recent connections, in the multivariate case, to the hard core model in statistical physics [15]). On the nature of these roots, Chudnovsky and Seymour [7] showed that the independence roots of clawfree graphs are all real, and Brown and Nowakowski [5] showed that with probability tending to 1, a graph will have a nonreal independence root.Asking when the independence roots are all real is a very natural question, but what about their location in the complex plane? While Brown et al. [4] showed that the collection of the independence roots of all graphs are in fact dense in the complex plane, plots of the independence roots of small graphs show a different story (see Figures 1 and 2). One striking thing about these plots is that not a single root lies in the open right half-plane (RHP) {z ∈ C : Re(z) > 0}, so we are left to wonder: how ubiquitous are graphs with stable independence polynomials, that is, with all their independence roots in the left half-plane (LHP) {z ∈ C : Re(z) ≤ 0}? (A polynomial with all of its roots in the LHP is called Hurwitz quasi-stable, or simply stable, and such polynomials are important in many applied settings [6]). Such a region is a natural extension of the negative real axis, which plays such a dominant role in the Chudnovsky-Seymour result on claw-free graphs.We shall call a graph itself stable if its independence polynomial is stable. It is known that the independence root of smallest modulus is always real and therefore negative (see [2]), so no independence polynomial has all its roots in the RHP, but it is certainly possible for it to have all roots in the LHP. This paper shall consider the stability of independence polynomials, providing some families of graphs whose independence polynomials are indeed stable, while showing that graphs formed under various constructions have independence polynomials that are not only nonstable but have roots with arbitrarily large real part.
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We bound the maximum modulus, maxmod(n), of an independence root over all graphs on n vertices and the maximum modulus, maxmod T (n), of an independence root over all trees on n vertices in terms of n. In particular, we show
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