### Basic Research Project J1-3001

Terwilliger algebra of a graph

Project Title: Terwilliger algebra of a graph

PI: Štefko Miklavič.

Project Code: J1-3001

Type of the Project: basic research project.

Funding Organization: Slovenian Research Agency (ARRS).

Research Field (ARRS): 1.01.00 - Natural Sciences and Mathematics / Mathematics.

Duration: 1. 10. 2021 - 30. 9. 2024.

Project Category: B.

Yearly Range: 1.57 FTE (2.667 research hours).

Sicris profile of the project is avaliable __here__.

Research Organizations:

University of Primorska, Andrej Marušič Institute,

University of Ljubljana, Faculty of Education.

Project Members:

Marušič Dragan (ARRS code: 02887)

Dobson Edward Tauscher (ARRS code: 34109)

Fernandez Blas (ARRS code: 52892)

Kovacs Istvan (ARRS code: 25997)

Kutnar Klavdija (ARRS code: 24997)

Malnič Aleksander (ARRS code: 02507)

Miklavič Štefko (leader)(ARRS code: 21656)

Milanič Martin (ARRS code: 30211)

Morgan Luke (ARRS code: 52908)

Pivač Nevena (ARRS code: 50673)

Šparl Primož (ARRS code: 23341)

Velkavrh Žiga (ARRS code: 50720)

Woodroofe Russ (ARRS code: 50355)

Abstract:

Our research concerns a combinatorial object known as a graph. A graph is a nite set of vertices, together with a set of undirected arcs or edges, each of which connects a pair of distinct vertices. We say that vertices x, y are adjacent whenever x, y are connected by an edge. The concept of a graph is useful because mathematical as well as intuitive notions can be formulated in terms of adjacency.

In our research the following situation occurs often. Let G be a graph and let H be a certain algebraic object, associated with G. In this case, one of the main motivations in our research is the following question: what could we say about the combinatorial properties of G, if we know that H has certain algebraic properties? And vice-versa: what could we say about the algebraic properties of H, if we know that G has certain combinatorial properties? Perhaps the most well-known example of this interplay between combinatorics and algebra is obtained if H is the automorphism group of G. In this case there are many relations between combinatorial properties of G and algebraic properties of H. For example, if H acts transitively on the set of vertices of G, then G is regular (in the sense that every vertex of G has the same number of neighbours). If we further know that the stabilizer H_x of a vertex x has exactly three orbits, then G is strongly regular. There are many more examples of this interplay available in the literature. In this project the algebraic object, associated with G, will not be its automorphism group, but rather a certain matrix algebra, called a Terwilliger algebra of a graph G. The main motivation, however, remains the same: what could we say about the combinatorial properties of G, if we know that its Terwilliger algebra has certain algebraic properties? And vice-versa: what could we say about the algebraic properties of Terwilliger algebra of G, if we know that G has certain combinatorial properties?

Let G denote a connected graph with a vertex set X. Let M_X denote the C-algebra of matrices with entries in complex numbers and with rows and columns indexed by X. Let V denote the vector space over complex numbers consisting of column vectors with entries in complex numbers and rows indexed by X. We observe M_X acts on V by left multiplication. Let A denote the adjacency matrix of G and observe that A is an element of M_X. The adjacency algebra M of G is a subalgebra of M_X, generated by the adjacency matrix A.

Fix a vertex x of G and let d denote its eccentricity. For each integer i between 0 and d let E*_i denote the diagonal matrix in M_X with (y,y)-entry 1 if the distance between x and y equals i, and 0 otherwise. Matrices E*_0, E*_1, ..., E*_d are called dual idempotents of G with respect to x. Dual idempotents form a basis for a commutative subalgebra M* of M_X. The subalgebra M* is called the dual adjacency algebra of G with respect to x.

Let T = T(x) denote the subalgebra of M_X generated by M, M*. We call T the Terwilliger algebra of G with respect to x. Algebra T was first defined in [P. Terwilliger, The subconstituent algebra of an association scheme I, J. Algebraic Combin., 1 (1992), 363-388, Definition 3.3]. It is mainly used to study distance-regular graphs. There are not a lot of results about Terwilliger algebra of graphs, which are not distance-regular. The proposed project aims to make an important step forward regarding the problems connected with Terwilliger algebras of graphs which ARE NOT distance-regular. Our research will be concentrated around graphs which are similar to distance-regular graphs. These are above all the class of so-called distance-biregular graphs and the class of so-called distance-semiregular graphs. We expect that the results obtained in the framework of this project will have a positive spillover effect beyond graph theory and mathematics. Our goal will also be to increase the visibility and the international impact of Slovene scientific results, by supporting young researchers to set up their own independent research teams.