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It is known that with a non-unit Pisot substitution one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization...
Alon and Tarsi presented in a previous paper a certain weighted sum over the set of all proper -colorings of a graph, which can be computed from its graph polynomial. The subject of this paper is another combinatorial interpretation of the same quantity, expressed in terms of the numbers of certain modulo flows in the graph. Some relations between graph parameters can be obtained by combining these two formulas. For example: The number of proper 3-colorings of a 4-regular graph and the number...
The Gutman index and the edge-Wiener index have been extensively investigated particularly in the last decade. An important stream of re- search on graph indices is to bound indices in terms of the order and other parameters of given graph. In this paper we present asymptotically sharp upper bounds on the Gutman index and the edge-Wiener index for graphs of given order and vertex-connectivity κ, where κ is a constant. Our results substantially generalize and extend known results in the area.
If is a connected graph of order , then by a hamiltonian coloring of we mean a mapping of into the set of all positive integers such that (where denotes the length of a longest path in ) for all distinct . Let be a connected graph. By the hamiltonian chromatic number of we mean
where the minimum is taken over all hamiltonian colorings of . The main result of this paper can be formulated as follows: Let be a connected graph of order . Assume that there exists a subgraph...
For a connected graph G with at least two vertices and S a subset of vertices, the convex hull is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with . Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs G and H...
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