Independence polynomials of k-tree related graphs

Authors:
Lanzhen Song;William Staton;Bing Wei
Affiliations:
Department of Mathematics, University of Mississippi, University, MS 38677, USA;Department of Mathematics, University of Mississippi, University, MS 38677, USA;Department of Mathematics, University of Mississippi, University, MS 38677, USA
Venue:
Discrete Applied Mathematics
Year:
2010

Citing 3
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Abstract

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let @a(G) denote the cardinality of a maximum independent set and f"s(G) for 0@?s@?@a(G) denote the number of independent sets of s vertices. The independence polynomial I(G;x)=@?"i"="0^@a^(^G^)f"s(G)x^s defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients f"s(T) for trees T with n vertices: (n+1-ss)@?f"s(T)@?(n-1s) for s=2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before.