Page 289 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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GRAPH POLYNOMIALS (MS-62)
dynamical systems on various networks. The asymptotics for finite compact metric non-directed
graphs was constructed earlier by V.L. Chernyshev, in collaboration with A.I.Shafarevich and
A.A. Tolchennikov. A polynomial that asymptotically approximates the number of possible
final positions of a random walk on an undirected metric graph with incommensurable edges
was described with the help of multiple Barnes-Bernoulli polynomials (see [2] for details).
Recently the asymptotics was constructed for a certain class of strongly connected directed
graphs (see [3]). This class has been called one-way Sperner graphs. For them it was proved that
the asymptotics of the number of possible final positions of a random walk on a metric graph
grows polynomially. The degree of a polynomial is equal to the number of incommensurable
oriented cycles minus one. The leading coefficient is determined by the product of the lengths
of such cycles and the sum of the lengths of all the edges of the graph. Computer simulations
show that this result is valid not only for one-way Sperner graphs.
References
[1] Berkolaiko G., Kuchment P., Introduction to Quantum Graphs, Mathematical Surveys and
Monographs, V. 186 AMS, 2014.
[2] Chernyshev V.L., Tolchennikov A.A., The Second Term in the Asymptotics for the Number
of Points Moving Along a Metric Graph, Regular and Chaotic Dynamics, 2017, vol. 22,
no. 8, pp. 937–948.
[3] Chernyshev V.L., Tolchennikov A.A., Asymptotics of the Number of Endpoints of a Ran-
dom Walk on a Certain Class of Directed Metric Graphs. Russian Journal of Mathematical
Physics, 2021, 28, 2. In press.
Degree Deviation Measure of Graphs
Ali Ghalavand, ali797ghalavand@gmail.com
University of Kashan, Islamic Republic of Iran
Coauthor: Seyed Ali Reza Ashrafi Ghomroodi
Let G be a simple graph with n vertices and m edges. The degree deviation measure of G is
defined as s(G) = v∈V (G) |degG(v) − 2m |. The aim of this talk is to report a positive answer
n
to the Conjecture 4.2 of [J. A. de Oliveira, C. S. Oliveira, C. Justel and N. M. Maia de Abreu,
Measures of irregularity of graphs, Pesq. Oper. 33 (3) (2013) 383–398].
287
dynamical systems on various networks. The asymptotics for finite compact metric non-directed
graphs was constructed earlier by V.L. Chernyshev, in collaboration with A.I.Shafarevich and
A.A. Tolchennikov. A polynomial that asymptotically approximates the number of possible
final positions of a random walk on an undirected metric graph with incommensurable edges
was described with the help of multiple Barnes-Bernoulli polynomials (see [2] for details).
Recently the asymptotics was constructed for a certain class of strongly connected directed
graphs (see [3]). This class has been called one-way Sperner graphs. For them it was proved that
the asymptotics of the number of possible final positions of a random walk on a metric graph
grows polynomially. The degree of a polynomial is equal to the number of incommensurable
oriented cycles minus one. The leading coefficient is determined by the product of the lengths
of such cycles and the sum of the lengths of all the edges of the graph. Computer simulations
show that this result is valid not only for one-way Sperner graphs.
References
[1] Berkolaiko G., Kuchment P., Introduction to Quantum Graphs, Mathematical Surveys and
Monographs, V. 186 AMS, 2014.
[2] Chernyshev V.L., Tolchennikov A.A., The Second Term in the Asymptotics for the Number
of Points Moving Along a Metric Graph, Regular and Chaotic Dynamics, 2017, vol. 22,
no. 8, pp. 937–948.
[3] Chernyshev V.L., Tolchennikov A.A., Asymptotics of the Number of Endpoints of a Ran-
dom Walk on a Certain Class of Directed Metric Graphs. Russian Journal of Mathematical
Physics, 2021, 28, 2. In press.
Degree Deviation Measure of Graphs
Ali Ghalavand, ali797ghalavand@gmail.com
University of Kashan, Islamic Republic of Iran
Coauthor: Seyed Ali Reza Ashrafi Ghomroodi
Let G be a simple graph with n vertices and m edges. The degree deviation measure of G is
defined as s(G) = v∈V (G) |degG(v) − 2m |. The aim of this talk is to report a positive answer
n
to the Conjecture 4.2 of [J. A. de Oliveira, C. S. Oliveira, C. Justel and N. M. Maia de Abreu,
Measures of irregularity of graphs, Pesq. Oper. 33 (3) (2013) 383–398].
287
