A non-trivial automorphism g of a graph ? is called semiregular if the only power gi fixing a vertex is the identity mapping, and it is called quasi-semiregular if it fixes one vertex and the only power gi fixing another vertex is the identity mapping. In this paper, we prove that K4, the Petersen graph and the Coxeter graph are the only connected cubic arc-transitive graphs admitting a quasi-semiregular automorphism, and K5 is the only connected tetravalent 2-arc-transitive graph admitting a quasi-semiregular automorphism. It will also be shown that every connected tetravalent G-arc-transitive graph, where G is a solvable group containing a quasi-semiregular automorphism, is a normal Cayley graph of an abelian group of odd order.
COBISS.SI-ID: 1541113028
Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of the so-called five-valued spectra Boolean functions whose Walsh spectra take the values in {0, ±2 ?1 , ±2 ?2 }. Moreover, these design methods mainly regard the specification of these functions in their algebraic normal form (ANF) domain. In this paper, we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions. Both necessary and sufficient conditions on the Walsh support of these functions are given, which then connects their design (in the spectral domain) to a family of the so-called totally (non-overlap) disjoint spectra plateaued functions. We identify some suitable families of plateaued functions having this property, thus providing some generic methods in the spectral domain. Furthermore, we also provide an extensive analysis of their constructions in the ANF domain and provide several generic design methods. The importance of this class of functions is manifolded, where apart from being suitable for some cryptographic applications, we emphasize their property of being constituent functions in the so-called four-bent decomposition.
COBISS.SI-ID: 1541637060
Let $ p\colon \tilde {X} \rightarrow X$ be a regular covering projection of connected graphs, where $ {\mathrm{CT}}_{\mathcal P}$ denotes the group of covering transformations. Suppose that a group $ G \leq \mathrm{Aut} \,X$ lifts along $\mathcal P$ to a group $ \tilde {G} \leq \mathrm{Aut} \,\tilde {X}$. The corresponding short exact sequence $ \mathrm{id} \rightarrow \mathrm {CT}_{\mathcal P} \rightarrow \tilde {G} \rightarrow G \rightarrow \mathrm{id}$ is split sectional over a $ G$-invariant subset of vertices $ \Omega \subseteq V(X)$ if there exists a sectional complement, that is, a complement $ \overline {G}$ to $ \mathrm{CT}_{\mathcal P}$ with a $ \overline {G}$-invariant section $ \overline {\Omega } \subset V(\tilde {X})$ over $ \Omega $. Such lifts do not split just abstractly but also permutationally in the sense that they enable a nice combinatorial description. Sectional complements are characterized from several viewpoints. The connection between the number of sectional complements and invariant sections on one side, and the structure of the split extension itself on the other, is analyzed. In the case when $ \mathrm{CT}_{\mathcal P}$ is abelian and the covering projection is given implicitly in terms of a voltage assignment on the base graph $ X$, an efficient algorithm for testing whether $ \tilde {G}$ has a sectional complement is presented. Efficiency resides on avoiding explicit reconstruction of the covering graph and the lifted group.
COBISS.SI-ID: 1540135364