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We describe a new approach towards the systematic construction of finite groups up to isomorphism. This approach yields a practical algorithm for the construction of finite solvable groups up to isomorphism. We report on a GAP…

Subgraph isomorphism counting is known as #P-complete and requires exponential time to find the accurate solution. Utilizing representation learning has been shown as a promising direction to represent substructures and approximate the…

Learning to optimize is a rapidly growing area that aims to solve optimization problems or improve existing optimization algorithms using machine learning (ML). In particular, the graph neural network (GNN) is considered a suitable ML model…

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many…

We study a variant of the subgraph isomorphism problem that is of high interest to the quantum computing community. Our results give an algorithm to perform pattern matching in quantum circuits for many patterns simultaneously,…

The graph isomorphism problem looks deceptively simple, but although polynomial-time algorithms exist for certain types of graphs such as planar graphs and graphs with bounded degree or eigenvalue multiplicity, its complexity class is still…

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism,…

Let G be a finite p-solvable group and P a Sylow p-subgroup of G. Suppose that $\gamma_{l(p-1)}(P)\subseteq \gamma_r(P)^{p^s}$ for $l(p-1)<r+s(p-1)$, then the p-length is bounded by a function depending on l.

Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any…

We investigate the constant-depth circuit complexity of the Isomorphism Problem, Minimum Generating Set Problem (MGS), and Sub(quasi)group Membership Problem (Membership) for groups and quasigroups (=Latin squares), given as input in terms…

Suppose that the finite group $G=AB$ is a mutually permutable product of two subgroups $A$ and $B$. By using Sylow numbers of $A$ and $B$, we present some new bounds of the $p$-length $l_p(G)$ of a $p$-solvable group $G$ and the nilpotent…

In this paper we combine many of the standard and more recent algebraic techniques for testing isomorphism of finite groups (GpI) with combinatorial techniques that have typically been applied to Graph Isomorphism. In particular, we show…

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of…

We introduce the framework of Deep Weisfeiler Leman algorithms (DeepWL), which allows the design of purely combinatorial graph isomorphism tests that are more powerful than the well-known Weisfeiler-Leman algorithm. We prove that, as an…

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the…

For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent…

In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there…

The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs $G$ and $H$, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list…

In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these…

It's important to design polynomial time algorithms to test if two graphs are isomorphic at least for some special classes of graphs. An approach to this was presented by Eugene M. Luks(1981) in the work \textit{Isomorphism of Graphs of…