No abstract

The well-known Moore bound $M(k,g)$ serves as a universal lower bound for the order of $k$-regular graphs of girth $g$. The excess $e$ of a $k$-regular graph $G$ of girth $g$ and order $n$ is the difference between its order $n$ and the corresponding Moore bound, $e=n - M(k,g) $. We find infinite families of parameters $(k,g)$, $g$ even, for which we show that the excess of any $k$-regular graph of girth $g$ is larger than $4$. This yields new improved lower bounds on the order of $k$-regular graphs of girth $g$ of smallest possible order; the so-called $(k,g)$-cages. We also show that the excess of the smallest $k$-regular graphs of girth $g$ can be arbitrarily large for a restricted family of $(k,g)$-graphs satisfying a very natural additional structural property.

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Generalized pseudostandard word u, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet A and by a directive sequence of involutory antimorphisms acting on A * . Prefixes of u with increasing length are constructed using pseudopalindromic closure operator. We show that

We use the description of the Schützenberger automata for amalgams of finite inverse semigroups given by Cherubini, Meakin, Piochi in [5] to obtain structural results for such amalgams. Schützenberger automata, in the case of amalgams of finite inverse semigroups, are automata with special structure possessing finite subgraphs, that contain all essential information about the automaton. Using this crucial fact, and the Bass-Serre theory, we show that the maximal subgroups of an amalgamated free-product are either isomorphic to certain subgroups of the original semigroups or can be described as fundamental groups of particular finite graphs of groups build from the maximal subgroups of the original semigroups.