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Register a new userObservations on cycles in a variant of the Collatz Graph
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It is well known that the Collatz Conjecture can be reinterpreted as the Collatz Graph with root vertex 1, asking whether all positive integers are within the tree generated. It is further known that any cycle in the Collatz Graph can be represented as a tuple, given that inputting them into a function outputs an odd positive integer; yet, it is an open question as to whether there exist any tuples not of the form $(2,2,...,2)$, thus disproving the Collatz Conjecture. In this paper, we explore a variant of the Collatz Graph, which allows the 3x+1 operation to be applied to both even and odd integers. We prove an analogous function for this variant, called the Loosened Collatz Function (LCF), and observe various properties of the LCF in relation to tuples and outputs. We then analyse data on the numbers that are in cycles and the length of tuples that represent circuits. We prove a certain underlying unique factorisation monoid structure for tuples to the LCF and provide a geometric interpretation of satisfying tuples in higher dimensions. Research into this variant of the Collatz Graph may provide reason as to why there exist no cycles in the Collatz Graph.
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The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture.
Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the Collatz conjecture based on bit representation of integers. The scheme neither encounters any cycles in the so called Collatz sequence and nor the sequence grows indefinitely. Experimental results show that the Collatz sequence starting at the given integer , oscillates for finite number of times, never exceeds 1.7 times (scaling factor) size of the starting integer and finally reaches the value 1. The experimental results show strong evidence that conjecture is correct and paves a way for theoretical proof.
Graph neural networks (GNNs) have been successfully employed in a myriad of applications involving graph-structured data. Theoretical findings establish that GNNs use nonlinear activation functions to create low-eigenvalue frequency content that can be processed in a stable manner by subsequent graph convolutional filters. However, the exact shape of the frequency content created by nonlinear functions is not known, and thus, it cannot be learned nor controlled. In this work, node-variant graph filters (NVGFs) are shown to be capable of creating frequency content and are thus used in lieu of nonlinear activation functions. This results in a novel GNN architecture that, although linear, is capable of creating frequency content as well. Furthermore, this new frequency content can be either designed or learned from data. In this way, the role of frequency creation is separated from the nonlinear nature of traditional GNNs. Extensive simulations are carried out to differentiate the contributions of frequency creation from those of the nonlinearity.
For a given graph $G(V,E)$ and one of its dominating set $S$, the subgraph $Gleft[Sright]$ induced by $S$ is a called a dominating tree if $Gleft[Sright]$ is a tree. Not all graphs has a dominating tree, we will show that a graph without cut vertices has at least one dominating tree. Analogously, if $Gleft[Sright]$ is a forest, then it is called a dominating forest. As special structures of graphs, dominating tree and dominating forest have many interesting application, and we will focus on its application on the problem of planar graph coloring.
The Bubble-sort graph $BS_n,,ngeqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set ${(1 2), (2 3),ldots, (n-1 n)}$. It is a bipartite graph containing all even cycles of length $ell$, where $4leqslant ellleqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,,ngeqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.
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