# Inverse problems in Computational Chemistry: The case of Inverse eigenvalue Problems for molecule structures

### Mathematics Background

An inverse eigenvalue problem (IEP) is the problem of reconstructing a matrix with a special structure from prescribed spectral data. By structure we mean the pattern of entries that are either zero or nonzero. There are many different types of IEPs with different level of difficulty which depends on the structure of the matrices which are to be reconstructed and the available eigen information. For example consider the following real Jacobi matrix:

$$A_n = \begin{bmatrix} a_1&b_1&0&0&0&0& \cdots & 0\\ b_1&a_2&b_2&0&0&0&\cdots&0\\ 0&b_2&a_3&b_3&0&0&\cdots&0\\ 0&0&b_3&a_4&b_4&0&\cdots&0\\ 0&0&0&b_4&a_5&b_5&\cdots&0\\ 0&0&0&0&\ddots&\ddots&\ddots&0\\ 0&0&0&0&\cdots&b_{n-2}&a_{n-1}&b_{n-1}\\ 0&0&0&0&\cdots&0&b_{n-1}&a_n \end{bmatrix}$$

Such that $$b_i$$s are nonzero and $$a_i$$s may or may not be zero. An IEP about this matrix is this:

Let $$A_i$$ be $$i$$th leading principle submatrix of $$A_n$$ and let $$n$$ be an even number, for a given Jacobi matrix $$J$$ of size $$n/2 \times n/2$$ and list of numbers $$\Lambda = \{\lambda_1, \cdots, \lambda_n\}$$, find an $$n \times n$$ Jacobi matrix $$A_n$$ such that $$J$$ is its minor principle submatrix and $$\Lambda$$ is its eigenvalues.

or

For a given list of real numbers $$\Lambda = \{\lambda_1, \cdots, \lambda_n \}$$ and $$\mu =\{ \mu_1, \cdots, \mu_{n-1} \}$$, find a Jacobi matrix $$A_n$$ such that $$\Lambda$$ is its eigenvalues and $$\mu$$ is eigenvalues of $$A_{n-1}$$.

Many different problems are considered by researchers in many papers.

Because graphs are appropriate tools for presenting a matrix structure, many papers concern IEP of graphs. For a given symmetric matrix $$A_{n \times n}$$, graph of $$A$$ is a graph $$G=(V,E)$$ such that $$V = \{v_1, \cdots,v_n\}$$ and edge $$\{v_i,v_j\} \in E$$ if entry $$[A]_{ij} \neq 0$$. In many papers IEP of matrix of graphs (path and broom graph (path graph is actually a symmetric Jacobi matrix), star graph, four different IEPs for paw graph and etc.) are investigated by researchers. Please note that it is also possible to consider IEP of a directed graph, in this case its matrix will not be symmetric anymore.

### What I am looking for ...

After a few years of studying IEPs in pure mathematics, I am looking for inverse eigenvalue problems with real practical application. many times, problems are modeled as a matrix and the limits on the problem are moved to the matrix entries or its spectral information and then researchers study the matrix. For example IEP of mass spring system is investigated which is a topic in physics.

IEP of problems in Computational Chemistry are rarely (or even never) investigated and because many entities and molecules in chemistry are look like some graphs (path or cycle), and graphs can also be modeled as a matrix, hence different types of IEPs may happen there. I ask for any help to introduce some field of studies or references or even cooperation to help me in doing this.