Number of walks and degree powers in a graph
نویسندگان
چکیده
This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees. Let G = (V,E) be a connected graph on n vertices, V = {1, 2, . . . , n}, with adjacency matrix A. For any integer k ≥ 1, let a ij denote the (i, j) entry of the power matrix A. Let D be the diagonal matrix with elements (D)ii = di (the degree of vertex i). Here we study the relationship between the sum of all walks of length k in G and the sum of the k-th powers of its degrees. As a main result, and answering in the affirmative a conjecture of Marc Noy [8], we will show that ∑
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عنوان ژورنال:
- Discrete Mathematics
دوره 309 شماره
صفحات -
تاریخ انتشار 2009