05C50 Online Seminar: Rajesh Kannan
- Date: 03/10/2023
- Time: 08:00
University of Manitoba
On the Eccentricity Matrices of Trees: Invertibility, Inertia and Spectral Symmetry
The eccentricity matrix ε(G) of a connected graph G is obtained from the distance matrix of G by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The eigenvalues of ε(G) are the ε-eigenvalues of G. It is well-known that the distance matrices of trees are invertible, and the determinant of such a matrix depend only on the number of vertices of the tree. We show that the eccentricity matrix of tree T is invertible if and only if either T is star or P_4. Also we show that any tree with odd diameter has 4 distinct ε-eigenvalues, and any tree with even diameter has the same number of positive and negative ε-eigenvalues (which is equal to the number of ’diametrically distinguished’ vertices). Finally, we will discuss trees with ε-eigenvalues that are symmetric with respect to the origin. This is joint work with Iswar Mahato.
The slides and a recording of this talk will be posted on the original website.
Stephen Kirkland, University of Manitoba
Hermie Monterde, University of Manitoba
The 05C50 Online is an international seminar about graphs and matrices held twice a month on Fridays.
Time: 8AM Pacific/10AM Central
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