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.