We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcovi's inertia bound are equal. This means that upsilon e(-) = m-(e(-)-k), where upsilon is the number of vertices, k is the regularity, e(-) is the smallest eigenvalue, and in is the multiplicity of e(-). We show that Delsarte cocliques do not exist for all Taylor's 2-graphs and for point graphs of generalized quadrangles of order (q, q(2)-q) for infinitely many q. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.

We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper *-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the recentlyintroduced multiplicative index of a unital channel. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which cannot be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum information theory.