Further Results on the Number of Walks in Graphs and Weighted Entry Sums of Matrix Powers

We consider the number of walks in undirected and directed graphs and, more generally, the weighted sum of entries of matrix powers. In this respect, we generalize an earlier result for Hermitian matrices. By using these inequalities for the entry sum of matrix powers, we deduce similar inequalities for iterated kernels. For further conceivable inequalities, we provide counterexamples in the form of graphs that contradict the corresponding statement for the number of walks. For the largest eigenvalue of adjacency matrices, we generalize a bound of Nikiforov that uses the number of walks. Furthermore, we relate the number of walks in graphs to the number of nodes and the number of edges in iterated directed line graphs.      «

We consider the number of walks in undirected and directed graphs and, more generally, the weighted sum of entries of matrix powers. In this respect, we generalize an earlier result for Hermitian matrices. By using these inequalities for the entry sum of matrix powers, we deduce similar inequalities for iterated kernels. For further conceivable inequalities, we provide counterexamples in the form of graphs that contradict the corresponding statement for the number of walks. For the...     »