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The geometry of non-unit Pisot substitutions

Milton Minervino, Jörg Thuswaldner (2014)

Annales de l’institut Fourier

It is known that with a non-unit Pisot substitution σ 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...

The graph polynomial and the number of proper vertex coloring

Michael Tarsi (1999)

Annales de l'institut Fourier

Alon and Tarsi presented in a previous paper a certain weighted sum over the set of all proper k -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 k 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 of Graphs with Given Vertex-Connectivity

Jaya Percival Mazorodze, Simon Mukwembi, Tomáš Vetrík (2016)

Discussiones Mathematicae Graph Theory

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.

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

If G is a connected graph of order n ≥ 1 , then by a hamiltonian coloring of G we mean a mapping c of V ( G ) into the set of all positive integers such that | c ( x ) - c ( y ) | ≥ n - 1 - D G ( x , y ) (where D G ( x , y ) denotes the length of a longest x - y path in G ) for all distinct x , y ∈ V ( G ) . Let G be a connected graph. By the hamiltonian chromatic number of G we mean min ( max ( c ( z ) ; z ∈ V ( G ) ) ) , where the minimum is taken over all hamiltonian colorings c of G . The main result of this paper can be formulated as follows: Let G be a connected graph of order n ≥ 3 . Assume that there exists a subgraph...

The hull number of strong product graphs

A.P. Santhakumaran, S.V. Ullas Chandran (2011)

Discussiones Mathematicae Graph Theory

For a connected graph G with at least two vertices and S a subset of vertices, the convex hull [ S ] G is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with [ S ] G = V ( G ) . 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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