Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.
Digraphs of maximum out-degree at most d > 1, diameter at most k > 1 and order N(d, k) = d + ... + d(k) are called almost Moore or (d, k)-digraphs. So far, the problem of their existence has been solved only when d = 2, 3 or k = 2, 3, 4. In this paper we derive the nonexistence of (d, k)-digraphs, with k > 4 and d > 3, under the assumption of a conjecture related to the factorization of the polynomials Phi(n)(1 + x + ... + x(k)), where Phi(n)(x) denotes the nth cyclotomic polynomial and 1 < n <= N(d, k). Moreover, we prove that almost Moore digraphs do not exist for the particular cases when k = 5 and d = 4, 5 or 6. (C) 2014 Elsevier Ltd. All rights reserved.Postprint (published version
A detailed study was carried out on enzymatic peeling of oranges in a reactor using an enzyme preparation. This work was focused on determining the changes that happen in the peel albedo of Navelina oranges. Thus, the conditions of temperature and concentration of the enzymatic preparation were optimized in order to produce the maximum weight loss, which indicates good peeling efficiency. Experiments to study the efficiency of the reactor effluents by reusing them for successive enzymatic peelings were also carried out with the aim of diminishing the cost of the process. Finally, the recovery of the enzymes after use in multiple enzymatic peelings by means of ultrafiltration was tested and an increment in their enzymatic activity was observed.
Regular digraphs of degree $d>1$, diameter $k>1$ and order $N(d,k) = d+\cdots +d^k$ will be called almost Moore $(d,k)$-digraphs. So far, the problem of their existence has only been solved when $d=2, 3$ or $k = 2, 3$. In this paper we prove that almost Moore digraphs of diameter 4 do not exist for any degree $d$.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.