progress

First time was spent on getting the basic notations, definitions on algebraic graph theory

Next up the author learn/understand/write the three different kinds of proof for the classic Hoffman bound
The proofs uses linear algebra, semi definiteness and interlacing of eigenvalues respectively

Next up he considered a graph generated from a combinatorial point of view, claim is that the largest independent set of this partition graph is an equality on Hoffman
more precisely its an equality for the smallest eigenvalue

the largest independent set and clique has been found using Erdos-Ko -Rado theorem and "Berrynice" theorem...inferring that this graph satisfies AlphaG.OmegaG <= V(G) at equality

the author tried to classify this partition graph under one of the few known regular graph that satisfy hoffman at equality but failed

So the attempt to prove the conjecture begins,
People from another universoty has suggested using association schemes, generalize the phenomanon to known group theory results and project it down again
The author is currently looking at the matrices and main graph and its equitable partition.
There seems a pattern with the "compression matrix" containing the largest and smallest lamda of its "elder sibling"
The author conjectures this more general fact and hopes to prove it, which by far will then be a result in matrix theory and yet serves to prove his original conjecture on combinatorial graph theory

Currently the author is trying to come up with random sample matrices for his compression theory and using MATLAB, the conjecture has not broken down yet.