we study the set of all determinants of adjacency matrices of graphs with a given number of vertices. using brendan mckay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. using an idea of m. newman, it is proved that if $...

Let S = {x1, x2, . . . , xn} be a set of distinct positive integers such that gcd(xi, xj) ∈ S for 1 ≤ i, j ≤ n. Such a set is called GCD-closed. In 1875/1876, H.J.S. Smith showed that, if the set S is “factor-closed”, then the determinant of the matrix eij = gcd(xi, xj) is det(E) = ∏n m=1 φ(xm), where φ denotes Euler’s Phi-function. Since the early 1990’s there has been a rebirth of interest in...

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.

Giant colonic diverticulum (GCD), defined as diverticulum larger than 4 cm, is a rare entity. It is generally a manifestation of colonic diverticular disease and can have dramatic complications such as perforation, abscess, volvulus, infarction and adenocarcinoma. This report documents the case of a 63-year-old man coming to the Emergency Department with acute abdomen due to a perforation of a ...

Let q be a prime power, Fq be the field of q elements, and k, m be positive integers. A bipartite graph G = Gq(g, h, k,m) is defined as follows. The vertex set of G is a union of two copies P and L of two-dimensional vector spaces over Fq, with two vertices (p1, p2) ∈ P and [ l1, l2 ] ∈ L being adjacent if and only if p2 + l2 = p1l 1 . We prove that graphs Gq(k, m) and Gq′(k,m) are isomorphic i...

Based on the Bezout approach we propose a simple algorithm to determine gcd of two polynomials which doesn’t need division, like Euclidean algorithm, or determinant calculations, Sylvester matrix algorithm. The needs only n steps for degree n. Formal manipulations give discriminant resultant any without needing division nor calculation.

0 binary, I-binary: The binary GCD algorithm ([lS]) and its improvement for multidigit integers (Gosper, see [12]). We compare the executron times of several algoixtliiiis for computing the G‘C‘U of arbitrary precasion iirlegers. These algorithms are the known ones (Euclidean, brnary, plus-mrnus), and the improved variants of these for multidigit compzltation (Lehmer and similar), as well as ne...

In recent work we called a ring R a GGCD ring if the semigroup of finitely generated faithful multiplication ideals of R is closed under intersection. In this paper we introduce the concept of generalized GCD modules. An R-moduleM is a GGCD module if M is multiplication and the set of finitely generated faithful multiplication submodules of M is closed under intersection. We show that a ring R ...

Mesial temporal sclerosis (MTS) is the most common lesion in chronic, intractable temporal lobe epilepsies (TLE) and characterized by segmental neuronal cell loss in major hippocampal segments. Another histopathological hallmark includes granule cell dispersion (GCD), an architectural disturbance of the dentate gyrus encountered in approximately 50% of patients with mesial temporal sclerosis. R...