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Register a new userRealization of digraphs in Abelian groups and its consequences
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Let $overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $Gamma$ be a finite Abelian group such that $|Gamma|geq 4|V(overrightarrow{G})|$ or if $|V(overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $varepsilon>0$ then $|Gamma|geq (1+varepsilon)|V(overrightarrow{G})|$. We show that there exists an injective mapping $varphi$ from $V(overrightarrow{G})$ to the group $Gamma$ such that $sum_{xin V(C)}varphi(x) = 0$ for every connected component $C$ of $overrightarrow{G}$, where $0$ is the identity element of $Gamma$. Moreover we show some applications of this result to group distance magic labelings.
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We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature in discrete spaces. An appealing feature of this discrete version seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest - particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (a la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs - a result of independent interest.
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $Tsubseteq G$, $Cay(G,S)cong Cay(G,T)$ implies that there is $sigmain Aut(G)$ such that $S^sigma=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if $8 mid n$, and is a NCI-group if and only if either $n=8$ or $8 mid n$.
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $alphain Aut(G)$ such that $S^alpha=T$, whenever $Cay(G,S)cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture proposed by Adam in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It takes about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order $2n$ is a NCI-group or a NDCI-group if and only if $n=2,4$ or $n$ is odd. As a direct consequence, we have that if a dihedral group $D_{2n}$ of order $2n$ is a DCI-group then $n=2$ or $n$ is odd-square-free, and that if $D_{2n}$ is a CI-group then $n=2,9$ or $n$ is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups.
A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=ab^{-1}$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the difference size of $G$ and is denoted by $Delta[G]$. The fraction $eth[G]:=frac{Delta[G]}{sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $pge 11$, any finite Abelian $p$-group $G$ has difference characteristic $eth[G]<frac{sqrt{p}-1}{sqrt{p}-3}cdotsup_{kinmathbb N}eth[C_{p^k}]<sqrt{2}cdotfrac{sqrt{p}-1}{sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.
Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sumsets.
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