An Example Usage of Graph Theory in Other Scientific Fields: On Graph Labeling, Possibilities and Role of Mind/Consciousness

This paper provides insights into some aspects of the possibilities and role of mind, consciousness, and their relation to mathematical logic with the application of problem solving in the fields of psychology and graph theory. This work aims to dispel certain long-held notions of a severe psychological disorder and a well-known graph labeling conjecture. The applications of graph labelings of various types for various kinds of graphs are being discussed. Certain results in graph labelings using computer software are presented with a direction to discover more applications

Strings And Colorings Of Topological Coding Towards Asymmetric Topology Cryptography

We, for anti-quantum computing, will discuss various number-based strings, such as number-based super-strings, parameterized strings, set-based strings, graph-based strings, integer-partitioned and integer-decomposed strings, Hanzi-based strings, as well as algebraic operations based on number-based strings. Moreover, we introduce number-based string-colorings, magic-constraint colorings, and vector-colorings and set-colorings related with strings. For the technique of encrypting the entire network at once, we propose graphic lattices related with number-based strings, Hanzi-graphic lattices, string groups, all-tree-graphic lattices. We study some topics of asymmetric topology cryptography, such as topological signatures, Key-pair graphs, Key-pair strings, one-encryption one-time and self-certification algorithms. Part of topological techniques and algorithms introduced here are closely related with NP-complete problems or NP-hard problems.Comment: Asymmetric topology encryption is a new topic of topological coding towards the certificateless public key cryptograph

On the graceful polynomials of a graph

Every graph can be associated with a family of homogeneous polynomials, one for every degree, having as many variables as the number of vertices. These polynomials are related to graceful labellings: a graceful polynomial with all even coefficients is a basic tool, in some cases, for proving that a graph is non-graceful, and for generating a possibly infinite class of non-graceful graphs. Graceful polynomials also seem interesting in their own right. In this paper we classify graphs whose graceful polynomial has all even coefficients, for small degrees up to 4. We also obtain some new examples of non-graceful graphs

Sailing towards, and then against, the Graceful Tree Conjecture: some promiscuous results

Vengono proposti argomenti a favore di una possibile, futura, risposta negativa alla congettura di Ringel sulla graziosità degli alberi. Viene fornita una classificazione delle etichettature per una sottoclasse elementare di alberi, sottolineando che l'informazione combinatoria crea ostruzioni algebriche (che tuttavia non pregiudicano la graziosità, almeno in questo caso)