# Combinatorics and Graph Theory

## 1805 Submissions

[2] **viXra:1805.0377 [pdf]**
*replaced on 2018-05-26 14:50:45*

### Envp, Another Prime Number Based Strategy to Encode Graphs

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

In this paper we show a method to encode graphs with a numerical value that follows unique labeling of each vertex or node and unique labeling of each edge of a graph with unique prime numbers. Each edge is defined as the connectivity between two vertices, therefore two vertices or nodes connected by an edge may be represented by the “ edge-nodes value ” derived by raising the prime number representing the edge to the product of the primes representing the two nodes that are connected by that edge. Multiplying all the “edge-nodes values” of a single graph will represent a unique number albeit very large in majority of cases. Given this unique number called the “Edge-nodes values product”, it is possible to derive the structure of the given graph. This encoding may allow new approaches to graph isomorphism, cryptography, quantum computing, data security, artificial intelligence, etc.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1805.0205 [pdf]**
*submitted on 2018-05-10 10:33:08*

### Labeled Trees with Fixed Node Label Sum

**Authors:** Richard J. Mathar

**Comments:** 70 Pages.

The non-cyclic graphs known as trees may be labeled by assigning
positive integer numbers (weights) to their vertices or to their edges.
We count the trees
up to 10 vertices that have prescribed sums of weights, or, from
the number-theoretic point of view, we count the compositions
of positive integers that are constrained by
the symmetries of trees.

**Category:** Combinatorics and Graph Theory