Spectra of Weighted Adjacency Matrices

Title: Spectra of Weighted Adjacency Matrices

Research Question: Can we develop a method to obtain lower bounds on the chromatic number using weighted adjacency matrices?

Methodology: The researchers introduced the concept of sign reversal, where they defined a map that changes the sign of a matrix while minimizing the cost. They used this concept to derive lower bounds on the cost in terms of the spectra of weighted adjacency matrices. These weighted adjacency matrices were constructed using the Hadamard product of the adjacency matrix and an arbitrary Hermitian matrix.

Results: They proved that for a connected graph G on n vertices, each diagonal element of the weighted adjacency matrix is positive, and the largest eigenvalue of the matrix is a lower bound for the chromatic number. They also showed that the cost of the sign reversal map can be reduced to χ−1, where χ is the chromatic number.

Implications: This research provides a new method to obtain lower bounds on the chromatic number using weighted adjacency matrices. It also contributes to the understanding of the spectra of weighted adjacency matrices and their relationship with the chromatic number. This can have implications in various fields such as graph theory, computer science, and quantum computing.

In summary, this research developed a new method to obtain lower bounds on the chromatic number using weighted adjacency matrices. This method involves the concept of sign reversal and the use of the Hadamard product. The results provide valuable insights into the properties of weighted adjacency matrices and their application in graph theory.

Link to Article: https://arxiv.org/abs/0112023v1 Authors: arXiv ID: 0112023v1