Graphs Whose Independence Fractals are Line Segments

Abstract

Let G be a simple graph. By an independent set in G, we mean a set of pairwise non-adjacent vertices in G. The independence polynomial of G is defined as $I_G(z)=i_0 + i_1 z + i_2 z^2+\cdots +i_\alpha z^{\alpha }$, where $i_m=i_m(G)$ is the number of independent sets in G with cardinality m and $\alpha =\alpha (G)$ denotes the cardinality of a largest independent set in G (known as the independence number of G). Let $G^k$ denote the k-times lexicographic product of G with itself. The set of roots of $I_{G^k}$ is known to converge as k tends to $\infty$, with respect to the Hausdorff metric, and the limiting set is known as the independence attractor. The independence fractal of a graph is the limiting set of roots of the reduced independence polynomial $I_{G^k}-1$ of $G^k$ as k tends to $\infty$. In this article, we consider the independence fractals of graphs with independence number 3. We attempt to find all such graphs whose independence fractal is a line segment. It is shown that the independence fractal and the independence attractor coincide when the earlier is a line segment. The line segment turns out to be an interval $[-\frac{4}{k}, 0]$ for $k \in \{1, 2, 3, 4\}$. It is found that each of these graphs have 9 vertices and there are exactly 13 such disconnected graphs. We show that there does not exist any connected graph for $k=4$. For $k=1$, there are 17 such connected graphs and for $k=2,3$ the number is quite large.

Appendix

By a set of graphs we mean here a set of nonisomorphic graphs. Thus, $\cup _I$ denote union upto isomorphism such that for sets of graphs X and Y, $X \cup _I Y$ yields a set of nonisomophic graphs from X and Y.