By Kuczma M.E. (ed.)

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**Extra info for 144 problems of the Austrian-Polish Mathematics Competition, 1978-1993**

**Sample text**

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 25 26 28 31 32 33 35 38 38 Introduction An independent set of a graph G is a set of pairwise nonadjacent vertices. The independence polynomial fG of G is the generating function of the sequence {fs }, where fs = fs (G) α is the number of independent vertex sets of cardinality s in G. That is, fG (x) = s=0 fs xs .

1 41 43 44 45 46 49 50 50 51 51 52 55 55 56 58 Introduction We will introduce the sandpile model by way of an explicit example. The sandpile model may be deﬁned on any reasonable graph, where by “reasonable” we will mean an undirected, connected, simple, and loop-free graph. 1, the graph G = (V, E) is deﬁned by its vertex set V = {v1 , v2 , v3 , v4 , v5 , v6 } and undirected edge set E = {{v1 , v2 }, {v1 , v3 }, {v1 , v4 }, {v2 , v3 }, {v2 , v4 }, {v2 , v5 }, {v3 , v5 }, {v3 , v6 }, {v4 , v5 }, {v5 , v6 }}.

Theor. Comput. Sci. 6: 69–90. , W. Staton, and B. Wei. 2013. A bound on the values of independence polynomials at −1/k for k-degenerate graphs. Discrete Math. 313: 1793–1798. , and F. Harary. 1970. On the corona of two graphs. Aequationes Math. 4: 322–324. Independence Polynomials of k-Trees and Compound Graphs 39 [5] Garey M. , and D. S. Johnson. 1979. Computers and intractability: A guide to the theory of NP-completeness. H. Freeman, New York. [6] Gutman I. 1992. Independence vertex sets in some compound graphs.