Inequalities for the extreme eigenvalues of block-partitioned Hermitian matrices with applications to spectral graph theory

Let $A=D_A+B$ be a block $r\times r$, $r\ge2$, Hermitian matrix of order $n$, where $D_A$ is the block diagonal part of $A$. The main results of the paper are Theorems 2.1 and 2.2, which state the sharp inequalities
$$ \lambda_1(A)\ge\lambda_1(D_A+\xi B)\quad\text{and}\quad\lambda_n(A)\le\lambda_n(D_A+\xi B),\qquad-\frac1{r-1}\le\xi\le1, $$
and analyze the equality cases. Some implications of these results are indicated. As applications, matrices occurring in spectral graph theory are considered, and new lower bounds on the chromatic number of a graph are obtained. Bibl. 7 titles.