Serbian Academy of Sciences and Arts, Serbia
- It is well known that the information on numbers of walks is contained in the powers of graph’s adjacency matrix. An ability to compare numbers of walks of arbitrary length between two graphs then implies inequalities between spectral radii and Estrada indices of those graphs. Comparisons of numbers of walks are usually obtained through injective embeddings of the set of walks of one graph into the set of walks of another graph, although some shortcuts are allowed from time to time. In the lecture we will review comparison methods appearing in literature, and specifically those used in recent results such as the proof of Belardo-Li Marzi-Simi\’c conjectured extension of the Li-Feng lemma, Andriantiana and Wagner’s result on greedy tree with given degree sequence, and ordering of starlike trees by the numbers of walks.