###### Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

## Summary

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes.

We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].

###### About Supergraphs. Part III

## Summary

The previous articles [5] and [6] introduced formalizations of the step-by-step operations we use to construct finite graphs by hand. That implicitly showed that any finite graph can be constructed from the trivial edgeless graph *K*
_{1} by applying a finite sequence of these basic operations. In this article that claim is proven explicitly with Mizar[4].

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Uniform Distribution of the Sequence of Balancing Numbers Modulo *m*

## Abstract

The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.