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Let G = (V, E) be a 2-connected plane graph on n vertices with outer face C such that every 2-vertex cut of G contains at least one vertex of C. Let P G (q) denote the chromatic polynomial of G. We show that (−1) n P G (q) > 0 for all 1 < q ≤ 1.2040.... This result is a corollary of a more general result that (−1) n Z G (q, w) > 0 for all 1 < q ≤ 1.2040...,… (More)

For any graph G, let W (G) be the set of vertices in G of degrees larger than 3. We show that for any bridgeless graph G, if W (G) is dominated by some component of G − W (G), then F (G, λ) has no roots in (1, 2), where F (G, λ) is the flow polynomial of G. This result generalizes the known result that F (G, λ) has no roots in (1, 2) whenever |W (G)| 2. We… (More)

For any positive integers a, b, c, d, let R a,b,c,d be the graph obtained from the complete graphs K a , K b , K c and K d by adding edges joining every vertex in K a and K c to every vertex in K b and K d. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of R a,b,c,d is either a real number or a… (More)

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