Let I(G;x) denote the independence polynomial of a graph G. In this paper we
study the unimodality properties of I(G; x) for some composite graphs G.
Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of
G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where
I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and
(a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is
log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is
unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is
unimodal. We also give two su_cient conditions when the independence
polynomial of a complete multipartite graph is unimodal or log-concave.
Finally, for every odd positive integer _ > 3, we find a connected graph G
not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real
zeros. This answers a problem of Mandrescu and Miric?a.
We discuss a new restricted m-ary overpartition function h m .n/, which is the number of hyper m-ary overpartitions of n, such that each power of m is allowed to be used at most m times as a non-overlined part. In this note we use generating function dissections to prove the following family of congruences for all n 0, m 4, j 0, 3 Ä k Ä m 1, and t 1:
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