Discrete Applied Mathematics Seminar by Himanshu Gupta: On the Rigenvalues of the Graphs $D(5,q)$

Speaker: Himanshu Gupta, PIMS postdoctoral fellow of mathematics and statistics, University of Regina

Title: On the Rigenvalues of the Graphs $D(5,q)$

Abstract: In 1995, Lazebnik and Ustimenko introduced the family of $q$-regular graphs $D(k,q)$, which is defined for any positive integer $k$ and prime power $q$. The connected components of the graph $D(k,q)$ have provided the best-known general lower bound on the size of a graph for any given order and girth to this day. Furthermore, Ustimenko conjectured that the second largest eigenvalue of $D(k,q)$ is always less than or equal to $2\sqrt{q}$, indicating that the graphs $D(k,q)$ are almost Ramanujan graphs. In this talk, we will discuss some recent progress on this conjecture. This includes the result that the second largest eigenvalue of $D(5,q)$ is less than or equal to $2\sqrt{q}$ when $q$ is an odd prime power.

This is joint work with Vladislav Taranchuk.

Discrete Applied Math Seminar

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