The independence polynomial of a graph G is the generating polynomial corresponding to its independent sets of different sizes. More formally, if a_k(G) denotes the number of independent sets of G of size k then I(G,z) := ∑_k (-1)^k a_k(G) z^k. The study of evaluating I(G,z) has several deep connections to problems in combinatorics, complexity theory and statistical physics. Consequently, the roots of the independence polynomial have been studied in detail. In particular, many works have provided regions in the complex plane that are devoid of any roots of the polynomial. One of the first such results showed a lower bound on the absolute value of the smallest root β(G) of the polynomial. Furthermore, when G is connected, Goldwurm and Santini established that β(G) is a simple real root of I(G,z) smaller than one. An alternative proof was given by Csikvári. Both proofs do not provide a gap from β(G) to the smallest absolute value amongst all the other roots of I(G,z). In this paper, we quantify this gap.

Abstract Logarithmic space-bounded complexity classes such as $$\textbf{L} $$ L and $$\textbf{NL} $$ NL play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators $$\textbf{para}_{\textbf{W}}$$ paraW and $$\textbf{para}_\beta $$ paraβ for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators $$\textbf{para}_{\textbf{W}}$$ paraW and $$\textbf{para}_\beta $$ paraβ by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, $$\textbf{para}_{{\textbf{W}}[1]}$$ paraW[1] and $$\textbf{para}_{\beta {\textbf{tail}}}$$ paraβtail . Then, we consider counting versions of all four operators and apply them to the class $$\textbf{L} $$ L . We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ #paraβtailL -hard and can be written as the difference of two functions in $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ #paraβtailL . These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of $$\#\textbf{para}_{\beta {\textbf{tail}}}\textbf{L} $$ #paraβtailL under parameterised logspace parsimonious reductions coincides with $$\#\textbf{para}_\beta \textbf{L} $$ #paraβL . In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.