ILAS 2022

ILAS 2022 Meeting: Invited Mini-Symposium (MS-8) on
Distance matrices of graphs

June 20–24, 2022, National University of Ireland, Galway

Last modified on Jun 24, 2022

This is a local page for the invited Minisymposium on "Distance matrices of graphs", at the 2022 meeting of the International Linear Algebra Society (ILAS). The Minisymposium takes place on June 21 (Tue) and 23 (Thu). The organizers are: Projesh Nath Choudhury and Apoorva Khare.

For information on hotels, and registration, click on the ILAS 2022 logo at the top of this page.

Distance matrices associated to graphs have been explored intensively in the literature for several decades now, both from an algebraic and a spectral viewpoint. They have connections to graph embeddings, communications networks, and quantum chemistry among other areas. This minisymposium will bring together researchers working on distance matrices from a variety of perspectives, and discuss modern approaches and recent results.

Aida Abiad     Projesh Nath
Choudhury     Carlos Alfaro

Leslie Hogben     Carolyn Reinhart


21 Jun (Tue),   10:30 Aida Abiad
21 Jun (Tue),   11:00 Projesh Nath Choudhury
21 Jun (Tue),   11:30 Carlos Alejandro Alfaro Montufar
23 Jun (Thu),   10:30 Leslie Hogben
23 Jun (Thu),   11:00 Carolyn L. Reinhart

Audience 1     MiniSymposium 8     Audience 2

Titles and Abstracts

Aida Abiad, Eindhoven University of Technology (Netherlands) and Ghent University (Belgium) 21 Jun (Tue),   10:30
Extending a conjecture of Graham and Lovász on the distance characteristic polynomial Venue: AC202

Abstract. Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree is unimodal with the maximum value occurring at $\lfloor \frac{n}{2} \rfloor$ for a tree $T$ of order $n$. We extend this old conjecture to block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of block graphs.

This is joint work with B. Brimkov, S. Hayat, A. Khramova and J. Koolen.

Projesh Nath Choudhury, Indian Institute of Science (India) 21 Jun (Tue),   11:00
Blowup-polynomials of graphs Venue: AC202

Abstract. Given a finite simple connected graph $G=(V,E)$ (or even a finite metric space), we introduce a novel invariant which we call its blowup-polynomial $p_G(n_v : v \in V)$. To do so, we compute the determinant of the distance matrix of the graph blowup, obtained by taking $n_v$ copies of the vertex $v$, and remove an exponential factor. First: we show that as a function of the sizes $n_v$, $p_G$ is a polynomial, is multi-affine, and is real-stable. Second: we show that the multivariate polynomial $p_G$ fully recovers $G$. Third: we obtain a novel characterization of the complete multi-partite graphs, as precisely those whose "homogenized" blowup-polynomials are Lorentzian/strongly Rayleigh. (Joint with Apoorva Khare.)

Carlos Alejandro Alfaro Montufar, Banco de México (Mexico) 21 Jun (Tue),   11:30
Distance ideals of graphs Venue: AC202

Abstract. Distance ideals of graphs generalize, among other graph parameters, the spectrum and the Smith normal form (SNF) of distance and distance Laplacian matrices. In particular, they allow us to introduce the notion of codeterminantal graphs, which generalize the concepts of cospectral and coinvariant graphs. We show computational results on codeterminantal graphs up to 9 vertices. Although the spectrum of several graph matrices has been widely used to determine graphs, the computational results suggest that the SNF of the distance Laplacian matrix seems to perform better for determining graphs. Finally, we show that complete graphs and star graphs are determined by the SNF of its distance Laplacian matrix.

This is joint work with Aida Abiad (Eindhoven University of Technology and Ghent University), Kristin Heysse (Macalester College), Libby Taylor (Stanford University) and Marcos C. Vargas (Banco de México).

Leslie Hogben, American Institute of Mathematics and Iowa State University (USA) 23 Jun (Thu),   10:30
Spectra of Variants of Distance Matrices of Graphs Venue: AC202

Abstract. In the last ten years, variants of the distance matrix of a graph, such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph, have been studied. This talk compares and contrasts techniques and results for these four variants of distance matrices. New results are obtained by cross-applying techniques from one variant of the distance matrix to another are presented.

This is joint work with Carolyn Reinhart (Swarthmore).

Carolyn L. Reinhart, Swarthmore College (USA) 23 Jun (Thu),   11:00
The distance matrix and its variants for digraphs Venue: AC202

Abstract. A directed graph, or digraph, is a graph in which edges are replaced by directional arcs. While the distance matrix and its variants are symmetric matrices when defined on graphs, these matrices are not necessarily symmetric on digraphs. Thus, some of the techniques used in the graph case no longer apply. This talk will discuss techniques used to study distance matrices for digraphs and some results they have yielded. New results regarding cospectrality for the distance matrix of digraphs will also be presented.

This is joint work with Leslie Hogben (Iowa State University and AIM).