05C50 Online Seminar: Rajesh Kannan

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.