finite graph
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2022 ◽  
Vol 22 (1&2) ◽  
pp. 38-52
Author(s):  
Ayaka Ishikawa ◽  
Norio Konno

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.


2021 ◽  
Author(s):  
◽  
Ellis Dawson

<p>We investigate strongly graded C*-algebras. We focus on graph C*-algebras and explore the connection between graph C*-algebras and Leavitt path algebras, both of which are $\Z$-graded. It is known that a graphical condition called \emph{Condition (Y)} is necessary and sufficient for Leavitt path algebras to be strongly graded. In this thesis we prove this can be translated to the graph C*-algebra and prove that a graph C*-algebra associated to a row-finite graph is strongly graded if and only if Condition (Y) holds.</p>


2021 ◽  
Author(s):  
◽  
Ellis Dawson

<p>We investigate strongly graded C*-algebras. We focus on graph C*-algebras and explore the connection between graph C*-algebras and Leavitt path algebras, both of which are $\Z$-graded. It is known that a graphical condition called \emph{Condition (Y)} is necessary and sufficient for Leavitt path algebras to be strongly graded. In this thesis we prove this can be translated to the graph C*-algebra and prove that a graph C*-algebra associated to a row-finite graph is strongly graded if and only if Condition (Y) holds.</p>


2021 ◽  
pp. 1-31
Author(s):  
T. Banica ◽  
J.P. McCarthy

Abstract A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting, the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.


Author(s):  
Roozbeh Hazrat ◽  
Lia Vaš

If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text]


2021 ◽  
Vol 76 (5) ◽  
pp. 821-881
Author(s):  
L. S. Efremova ◽  
E. N. Makhrova

Abstract The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.


2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Anuwar Kadir Abdul Gafur ◽  
Suhadi Wido Saputro

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W   ⊆   V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .


Author(s):  
Daniel Gonçalves ◽  
Danilo Royer

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition (L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.


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