First half an hour (for Pre-seminar talk) I will try to introduce (and prove) some basic definitions and claims from Algebraic graph theory i.e. 1. What is a graph 2. Some partition around a vertex 3. What are primitive idempotents 4. What is the local spectrum of a vertex 5. The local scalar product of a vertex.

Main part of my talk is about algebraic characterizations around a vertex x and combinatorial properties (around x) of (non)regular graphs in case when one specific restriction holds. I am interested in two things: (i) what combinatorial properties of graphs can can tell us about T-modules of a graph; (ii) what algebraic properties of T-modules can tell us about combinatorial properties of a graph. Just to remind a reader that by my definition, T-module is a vector subspace W such that TW\$\subseteq\$W, where T is algebra of matrices (known as Terwilliger algebra) and these matrices are defined using the structure of a graph.