## The PIMS Postdoctoral Fellow Seminar: Mahsa N Shirazi

- Date: 02/08/2023

Online

Extensions of the Friendship Theorem

**Abstract:** For r ≥ 1, a graph has r-friendship property if every pair of vertices has exactly r common neighbours. The motivation for this definition is from the friendship theorem, which is on the graphs with 1-friendship property. The friendship theorem, first proved by Erdös, Rényi, and Sós in 1996, states that if G is a graph in which every pair of vertices has exactly one common neighbour, then G has a universal vertex v adjacent to all others, and the graph induced by V (G) \ {v} is a matching.

In this talk, we present a brief history of the problem, we study graphs with r-friendship property, where r ≥ 2. We show all such graphs are strongly regular. Furthermore, we prove that for any r ≥ 2, there are only finitely many graphs with r-friendship property. We provide some classes of strongly regular graphs with r-friendship property, and their connections to design theory. At the end, we discuss about some open problems and con- jectures on this topic.

This is an ongoing joint work with Karen Gunderson.

**Speaker biography:** Mahsa N Shirazi is a PIMS-CRG postdoctoral fellow at the University of Manitoba, under the supervision of Karen Gunderson. Her research is centered around using graph theory to answer questions that arise in design theory. Her current research project is on a special type of strongly regular graphs called r-friendship graphs. She is also interested in EKR problems on different mathematical objects, like perfect matchings and uniform set-partitions. She is also interested in Matrix theory and linear algebra.

She completed her Ph.D. at the University of Regina under the supervision of Shaun Fallat and Karen Meagher. Her main Ph.D. research was on the extensions of the famous Erdös-Ko-Rado theorem to t-intersecting families of perfect matchings in graphs. She obtained her master's degree in numerical linear algebra at the Shiraz University of Technology, under the supervision of Behnam Hashemi. Besides research and studying, she also enjoys painting and baking.

**Medium**: Read more about Mahsa and their research here.

This event is part of the Emergent Research: The PIMS Postdoctoral Fellow Colloquium Series.

This seminar takes places across multiple time zones: 9:30 AM Pacific/ 10:30 AM Mountain / 11:30 AM Central

Register via Zoom to receive the link for this event and the rest of the series.

See past seminar recordings on MathTube.