# THE SPECTRAL ANALYSIS OF RANDOM GRAPH MATRICES

In this thesis random graph models are under study. This is a set of graphs together with a probability distribution on that set. A random matrix is a matrix with entries consisting of random values from some specified distribution.

‘In daily life, as well as in industry and human behaviour, complex relations exist between numbers containing a direction,’ Dan Hu explains. ‘Visiting websites is a good example: we go from one site to the other quite randomly. When modelling links and edges, we should be able to add random graphs. The values are no longer only real here, but are complex in nature.’

The spectra of the corresponding matrices are called the spectra of the random graph. This thesis contains a number of results on the spectra, and related spectral properties, of several random graph models.

The main body of the thesis concerns the estimation of the eigenvalues of the Laplacian matrix of random multipartite graphs. Some spectral properties, such as the Laplacian energy, the Laplacian Estrada index, and the von Neumann entropy, were explored and discussed in this thesis work.

## NEW GRAPH MODEL

The second part discusses Guo’s and Mohar’s mixed graphs. ‘We proposed a new random graph model – the random mixed graph,’ Dan says. ‘In an early stage of my PhD project already, we proved that the empirical spectral distribution of the eigenvalues of the Hermitian adjacency matrix converges to Wigner’s semicircle law. Here, we used other mathematical techniques than were done ever before, in order to prove that with “weaker conditions” the same results holds as in Wigner’s law. I was very excited about this findings, and it led to a good publication.’

## OTHER RESULTS

Strengthened by this result, Dan dealt with other applications and topics within this mathematical field of research, leading to in total seven publications.

‘We provided several results on the spectra of general random mixed graphs,’ Dan states. ‘In particular, we presented a new probability inequality for sums of independent, random, self-adjoint matrices, and then applied this probability inequality to matrices arising from the study of random mixed graphs.’

## FUTURE WORK

Dan worked on this thesis project for one year in Twente, at the Discrete Mathematics and Mathematical Programming Group. ‘Professor Hajo Broersma is an expert in this field, and we had many discussions which were crucial for my vast progress,’ she says. ‘In the near future I will return to Northwestern Polytechnical University in Xi’an, to work on the next phase of my double PhD degree.’

‘After that I favor an academic career in China, combining mathematical research and teaching. I would like to motivate young students to go and work on graphs theory. There are so many fascinating conjectures left to be solved and shown relevant. This work will be of importance or various areas in mathematics including probability theory, linear algebra and matrix analyses.’