It has been known that the graph isomorphism problem is polynomial-time many-one reducible to the ring isomorphism problem. In fact, two different reductions have already been proposed. For those reductions, rings of certain types have been used to represent a given graph. In this paper, we give yet another reduction, which is based on a simpler and more natural construction of a ring from a gr...

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G ∼= H . For several decades, the np n+O(1) generatorenumeration bound (where p is the smallest prime dividing the order of the group) has been the best worst-case result for general groups. In this work, we show the first improvement over the generator-enumeration boun...

background : the incidence of thyroid cancer has varied from 2 per 100,000 in europe to 21 per 100,000 in the hawaiian chinese population and is 2-3 fold more common in females. middle east cancer consortium figures from 1996-2001 have recorded different age standardized incidence rates that ranged from 2 per 100,000 in egypt to 7.5 per 100,000 among israeli jews. in jordan the age standardized...

Any linear transformation can be represented by its matrix representation. In an ideal situation, all linear operators can be represented by a diagonal matrix. However, in the real world, there exist many linear operators that are not diagonalizable. This gives rise to the need for developing a system to provide a beautiful matrix representation for a linear operator that is not diagonalizable....

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies e...

We extend the concept of graph isomorphisms to multilayer networks, and we identify multiple types of isomorphisms. For example, in multilayer networks with a single “aspect” (i.e., type of layering), permuting vertex labels, layer labels, and both of types of layers each yield a different type of isomorphism. We discuss how multilayer network isomorphisms naturally lead to defining isomorphism...

This theory provides a compact formulation of Gauss-Jordan elimination for matrices represented as functions. Its distinctive feature is succinctness. It is not meant for large computations. 1 Gauss-Jordan elimination algorithm theory Gauss-Jordan-Elim-Fun imports Main begin Matrices are functions: type-synonym ′a matrix = nat ⇒ nat ⇒ ′a In order to restrict to finite matrices, a matrix is usua...

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c ≥ 1 of deletion: 1. GI ≡iso VDCc, GI ≡iso EDCc, GI ≤m LVDc, and GI ≡iso LEDc. 2...

While in general it is not known whether there is a polynomial time algorithm to decide whether two given graphs are isomorphic, there are polynomial-time algorithms for certain subsets of graphs, including but not limited to planar graphs and graphs with bounded valence. In this thesis, we will give a brief introduction on the Graph Isomorphism Problem and its relations to complexity theory. W...