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In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $rho = 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$ and $2^m$, respectively, where $m$ is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families of distance transitive coset graphs of diameter three and four, respectively, are constructed.
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Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the supplementary code of the other one. We obtain that if one of these codes is a Hamming code, then the supplementary code is completely regular and completely transitive. If one of the codes is completely regular with covering radius $2$, then the supplementary code is also completely regular with covering radius at most $2$. Moreover, in this case, either both codes are completely transitive, or both are not. With this technique, we obtain infinite families of completely regular and completely transitive codes which are quasi-perfect uniformly packed.
In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $rho$ equal to $3$ or $4$, and are $1/2^i$-th parts, for $iin{1,ldots,u}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where $m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.
In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.
A graph $G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {em bi-Cayley graph/} over $H$. Such a graph $G$ is called {em normal/} if $H$ is normal in the full automorphism group of $G$, and {em normal edge-transitive/} if the normaliser of $H$ in the full automorphism group of $G$ is transitive on the edges of $G$. % In this paper, we give a characterisation of normal edge-transitive bi-Cayley graphs, %which form an important subfamily of bi-Cayley graphs, and in particular, we give a detailed description of $2$-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic $p$-groups. We find that under certain conditions, `normal edge-transitive is the same as `normal for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most $6$, and answer some open questions from the literature about $2$-arc-transitive, half-arc-transitive and semisymmetric graphs.
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. As an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990.
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