All Questions

Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively. Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...

In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have $$ \min_{Y \in Y^*} \text{tr}(Y^TAY) = \text{tr}(X^TAX) = \sum_{i=1}^p \lambda_i, $$ with $$ \text{ $X^...

Kindly help me to prove/disprove the following statement. Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...

This question is motivated by a talk I went to earlier today. Suppose we have a $d$-regular graph $G$ with $n$ vertices, with adjacency matrix $A$. Let $$\lambda_1\geq \lambda_2 \geq\dots \geq \...