Let a and b be positive, relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In...

In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.

The classical Frobenius–Schur indicators for finite groups are character sums defined for any representation and any integer m ≥ 2. In the familiar case m = 2, the Frobenius–Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bu...

A permutation with no fixed points is called a derangement.The subset $mathcal{D}$ of a permutation group is derangement if all elements of $mathcal{D}$ are derangement.Let $G$ be a permutation group, a derangementgraph is one with vertex set $G$ and derangement set $mathcal{D}$ as connecting set. In this paper, we determine the spectrum of derangement graphs of order a product of three primes.

In this paper, we consider the minimum Hamming weight for linear codes over special finite quasi-Frobenius rings. Furthermore, we obtain minimal free $R$-submodules of a finite quasi-Frobenius ring $R$  which contain a linear code and derive the relation between their minimum Hamming weights. Finally, we suggest an algorithm that computes this weight using the Grobner basis and we show that und...

The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...

In [Frobenius Categories Versus Brauer Blocks, Progress in Math., 274] and [Ordinary Grothendieck groups of a Frobenius [Formula: see text]-category, Algebra Colloq. 18 (2011) 1–76], we consider suitable inverse limits categories modules characteristics text] zero, obtained from folded text]-category[Formula: text], which covers the case Frobenius[Formula: text]-categories associated with block...