Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Unsolved Problems in Spectral Graph Theory

Abstract: Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.

On the symmetric doubly stochastic matrices that are determined by their spectra

Abstract: A symmetric doubly stochastic matrix A is said to be determined by its spectra if the only symmetric doubly stochastic matrices that are similar to A are of the form $P^TAP$ for some permutation matrix P. The problem of characterizing such matrices is considered here. An almost the same but a more difficult problem was proposed by [ M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Lin. Alg. Appl., 432 (2010) 2925-2927] as follows: Characterize all n-tuples $\lambda= (1,\lambda_2,...,\lambda_n)$ such that up to a permutation similarity, there exists a unique symmetric doubly stochastic matrix with spectrum $\lambda.$ In this short note, some general results concerning our two problems are first obtained. Then, we completely solve these two problems for the case n = 3. Some connections with spectral graph theory are then studied. Finally, concerning the general case, two open questions are posed and a conjecture is introduced.

Adjacency graph and matrix representation of scaling mechanisms

Abstract: Scaling mechanisms are widely used in practice because of their deployable and foldable characteristics. But the representation of the topological structure and evolution process of the mechanisms are little referred to. This paper is to introduce adjacency graph and its adjacency matrix operation, which are used to represent the topological structure and the evolution process of scaling mechanisms. In this paper, the method which can be extended to some other topological changed mechanisms represents the topological information intuitively and comprehensively.

Spectral Graph Theory Based Resource Allocation for IRS-Assisted Multi-Hop Edge Computing

Abstract: The performance of mobile edge computing (MEC) depends critically on the quality of the wireless channels. From this viewpoint, the recently advocated intelligent reflecting surface (IRS) technique that can proactively reconfigure wireless channels is anticipated to bring unprecedented performance gain to MEC. In this paper, the problem of network throughput optimization of an IRS-assisted multi-hop MEC network is investigated, in which the phase-shifts of the IRS and the resource allocation of the relays need to be jointly optimized. However, due to the coupling among the transmission links of different hops caused by the utilization of the IRS and the complicated multi-hop network topology, it is difficult to solve the considered problem by directly applying existing optimization techniques. Fortunately, by exploiting the underlying structure of the network topology and spectral graph theory, it is shown that the network throughput can be well approximated by the second smallest eigenvalue of the network Laplacian matrix. This key finding allows us to develop an effective iterative algorithm for solving the considered problem. Numerical simulations are performed to corroborate the effectiveness of the proposed scheme.