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Linearly bounded infinite graphs
Linearly bounded Turing machines have been mainly studied as acceptors for
context-sensitive languages. We define a natural class of infinite automata
representing their observable computational behavior, called linearly bounded
graphs. These automata naturally accept the same languages as the linearly
bounded machines defining them. We present some of their structural properties
as well as alternative characterizations in terms of rewriting systems and
context-sensitive transductions. Finally, we compare these graphs to rational
graphs, which are another class of automata accepting the context-sensitive
languages, and prove that in the bounded-degree case, rational graphs are a
strict sub-class of linearly bounded graphs
Weighted spanning trees on some self-similar graphs
We compute the complexity of two infinite families of finite graphs: the
Sierpi\'{n}ski graphs, which are finite approximations of the well-known
Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group
acting on the rooted ternary tree. For both of them, we study the
weighted generating functions of the spanning trees, associated with several
natural labellings of the edge sets.Comment: 21 page
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