## Asymptotic results regarding the number of walks in a graph

Dress A, Gutman I (2003)
Applied Mathematics Letters 16(3): 389-393.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

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Autor*in
Dress, AndreasUniBi; Gutman, Ivan
Abstract / Bemerkung
Let W-k denote the number of walks of length k(greater than or equal to 0) in a finite graph G, and define Delta(k) = Deltak(G) := Wk+1 Wk-1 - Wk. The condition A(2k-1) (G) > 0 is satisfied for all k is an element of N, and for all graphs G except the harmonic graphs for which Delta(k) = 0 holds for all k > 2-and thus, in particular, for the regular graphs for which Delta(k) = 0 holds for all k is an element of N. In contrast, the sign Of Delta(2k)(G) may be positive as well as negative, depending on k and G. We show that, for every finite graph G, there exist unique numbers N is an element of N-0, tau(1),...,tau(N), a(1),...,a(N), b(1),...,b(N) is an element of R-greater than or equal to0 with 0 less than or equal to tau(1) < tau(2) < ... < tau(N) and a(1) + b(1), a(2) + b(2),...,a(N) + b(N) > 0 that can be computed in terms of the main eigenvalues and -angles of G such that Delta(k) = Sigma(nu-1)(N)(a(nu) +(-1)(k-1) b(nu)) t(nu)(k-1) holds for all k in N. Consequently, the limits lim(k-->infinity) (2k)rootDelta(2k+1) and lim(k-->infinity) (2k-1)rootDelta(2k) always exist and the sign Of Delta(2k) is constant for all sufficiently large k. (C) 2003 Elsevier Science Ltd. All rights reserved.
Stichworte
graphs; harmonic; semiregular graphs; regular graphs; eigenvectors (of graphs); semiharmonic graphs; walks in graphs; spectral graph theory; eigenvalues (of graphs)
Erscheinungsjahr
2003
Zeitschriftentitel
Applied Mathematics Letters
Band
16
Ausgabe
3
Seite(n)
389-393
ISSN
0893-9659
Page URI
https://pub.uni-bielefeld.de/record/1612286

## Zitieren

Dress A, Gutman I. Asymptotic results regarding the number of walks in a graph. Applied Mathematics Letters. 2003;16(3):389-393.
Dress, A., & Gutman, I. (2003). Asymptotic results regarding the number of walks in a graph. Applied Mathematics Letters, 16(3), 389-393. https://doi.org/10.1016/S0893-9659(03)80062-9
Dress, Andreas, and Gutman, Ivan. 2003. “Asymptotic results regarding the number of walks in a graph”. Applied Mathematics Letters 16 (3): 389-393.
Dress, A., and Gutman, I. (2003). Asymptotic results regarding the number of walks in a graph. Applied Mathematics Letters 16, 389-393.
Dress, A., & Gutman, I., 2003. Asymptotic results regarding the number of walks in a graph. Applied Mathematics Letters, 16(3), p 389-393.
A. Dress and I. Gutman, “Asymptotic results regarding the number of walks in a graph”, Applied Mathematics Letters, vol. 16, 2003, pp. 389-393.
Dress, A., Gutman, I.: Asymptotic results regarding the number of walks in a graph. Applied Mathematics Letters. 16, 389-393 (2003).
Dress, Andreas, and Gutman, Ivan. “Asymptotic results regarding the number of walks in a graph”. Applied Mathematics Letters 16.3 (2003): 389-393.
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