Towards a conjecture on the distance spectral radius of trees
Valisoa Razanajatovo Misanantenaina
Minisymposium: SPECTRAL GRAPH THEORY
Content: The distance matrix of a graph is the matrix whose entry in the $i$th row, $j$th column is the distance $d(v_i,v_j)$ between the $i$th vertex $v_i$ and the $j$th vertex $v_j$. A conjecture of Ili\'c and Stevanovi\'c states that among all trees with given order and maximum degree, the so-called Volkmann trees minimise the spectral radius of the distance matrix. In this talk, we present our recent progress towards this conjecture and its analogue for a ``reversed'' version of the distance matrix. Our ideas are based on similar results for the Wiener index and the spectral radius of the adjacency matrix.