• Title/Summary/Keyword: finite graph

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ON THE ANNIHILATOR GRAPH OF GROUP RINGS

  • Afkhami, Mojgan;Khashyarmanesh, Kazem;Salehifar, Sepideh
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.331-342
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    • 2017
  • Let R be a commutative ring with nonzero identity and G be a nontrivial finite group. Also, let Z(R) be the set of zero-divisors of R and, for $a{\in}Z(R)$, let $ann(a)=\{r{\in}R{\mid}ra=0\}$. The annihilator graph of the group ring RG is defined as the graph AG(RG), whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices x and y are adjacent if and only if $ann(xy){\neq}ann(x){\cup}ann(y)$. In this paper, we study the annihilator graph associated to a group ring RG.

STRUCTURAL PROPERTIES FOR CERTAIN GLASSES OF INFINITE PLANAR GRAPHS

  • Jung, Hwan-Ok
    • Journal of applied mathematics & informatics
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    • v.13 no.1_2
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    • pp.105-115
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    • 2003
  • An infinite locally finite plane graph is called an LV-graph if it is 3-connected and VAP-free. If an LV-graph G contains no unbounded faces, then we say that G is a 3LV-graph. In this paper, a structure theorem for an LV-graph concerning the existence of a sequence of systems of paths exhausting the whole graph is presented. Combining this theorem with the early result of the author, we obtain a necessary and sufficient conditions for an infinite VAP-free planar graph to be a 3LV-graph as well as an LV-graph. These theorems generalize the characterization theorem of Thomassen for infinite triangulations.

INTERSECTION GRAPH에 관하여

  • Kim Yuon Sik
    • The Mathematical Education
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    • v.13 no.2
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    • pp.6-10
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    • 1974
  • We consider 'ordinary' graphs: that is, finite undirected graphs with no loops or multiple edges. An intersection representation of a graph G is a function r from V(G), the set of vertices of G, into a family of sets S such that distinct points $\chi$$_{\alpha}$ and $\chi$$_{\beta}$/ of V(G) are. neighbors in G precisely when ${\gamma}$($\chi$$_{\alpha}$)∩${\gamma}$($\chi$$_{\beta}$/)$\neq$ø, A graph G is a rigid circuit grouph if every cycle in G has at least one triangular chord in G. In this paper we consider the main theorem; A graph G has an intersection representation by arcs on an acyclic graph if and only if is a normal rigid circuit graph.uit graph.d circuit graph.uit graph.

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Characteristic polynomials of graph bundles with productive fibres

  • Kim, Hye-Kyung;Kim, Ju-Young
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.75-86
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    • 1996
  • Let G be a finite simple connected graph with vertex set V(G) and edge set E(G). Let A(G) be the adjacency matrix of G. The characteristic polynomial of G is the characteristic polynomial $\Phi(G;\lambda) = det(\lambda I - A(G))$ of A(G). A zero of $\Phi(G;\lambda)$ is called an eigenvalue of G.

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VARIATION OF ORBIT-COINCIDENCE SETS

  • Srivastava, Anjali
    • Journal of the Chungcheong Mathematical Society
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    • v.15 no.1
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    • pp.1-6
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    • 2002
  • David Gavid [3] proved that in many familiar cases the upper semi-finite topology on the set of closed subsets of a space is the largest topology making the coincidence function continuous, when the collection of functions is given the graph topology. Considering G-spaces and taking the coincidence set to consist of points where orbits coincidence, we obtain G-version of many of his results.

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ON RINGS WHOSE ANNIHILATING-IDEAL GRAPHS ARE BLOW-UPS OF A CLASS OF BOOLEAN GRAPHS

  • Guo, Jin;Wu, Tongsuo;Yu, Houyi
    • Journal of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.847-865
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    • 2017
  • For a finite or an infinite set X, let $2^X$ be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set $2^X{\setminus}\{X,{\emptyset}\}$, with M adjacent to N if $M{\cap}N={\emptyset}$. In this paper, we characterize the annihilating-ideal graphs $\mathbb{AG}(R)$ that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively. In particular, for a commutative ring R such that AG(R) has a maximum clique S with $3{\leq}{\mid}V(S){\mid}{\leq}{\infty}$, we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if R is a reduced ring. If assume further that R is decomposable, then we prove that $\mathbb{AG}(R)$ is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph $\mathbb{AG}(R)$.

LINE GRAPHS OF UNIT GRAPHS ASSOCIATED WITH THE DIRECT PRODUCT OF RINGS

  • Pirzada, S.;Altaf, Aaqib
    • Korean Journal of Mathematics
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    • v.30 no.1
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    • pp.53-60
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    • 2022
  • Let R be a finite commutative ring with non zero identity. The unit graph of R denoted by G(R) is the graph obtained by setting all the elements of R to be the vertices of a graph and two distinct vertices x and y are adjacent if and only if x + y ∈ U(R), where U(R) denotes the set of units of R. In this paper, we find the commutative rings R for which G(R) is a line graph. Also, we find the rings for which the complements of unit graphs are line graphs.

EXISTENCE OF GLOBAL SOLUTIONS TO SOME NONLINEAR EQUATIONS ON LOCALLY FINITE GRAPHS

  • Chang, Yanxun;Zhang, Xiaoxiao
    • Journal of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.703-722
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    • 2021
  • Let G = (V, E) be a connected locally finite and weighted graph, ∆p be the p-th graph Laplacian. Consider the p-th nonlinear equation -∆pu + h|u|p-2u = f(x, u) on G, where p > 2, h, f satisfy certain assumptions. Grigor'yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V. In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m-order differential operator 𝓛m,p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.

MULTIPLE SOLUTIONS OF A PERTURBED YAMABE-TYPE EQUATION ON GRAPH

  • Liu, Yang
    • Journal of the Korean Mathematical Society
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    • v.59 no.5
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    • pp.911-926
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    • 2022
  • Let u be a function on a locally finite graph G = (V, E) and Ω be a bounded subset of V. Let 𝜀 > 0, p > 2 and 0 ≤ λ < λ1(Ω) be constants, where λ1(Ω) is the first eigenvalue of the discrete Laplacian, and h : V → ℝ be a function satisfying h ≥ 0 and $h{\not\equiv}0$. We consider a perturbed Yamabe equation, say $$\{\begin{array}{lll}-{\Delta}u-{\lambda}u={\mid}u{\mid}^{p-2}u+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω and ∂Ω denote the interior and the boundary of Ω, respectively. Using variational methods, we prove that there exists some positive constant 𝜀0 > 0 such that for all 𝜀 ∈ (0, 𝜀0), the above equation has two distinct solutions. Moreover, we consider a more general nonlinear equation $$\{\begin{array}{lll}-{\Delta}u=f(u)+{\varepsilon}h,&&\text{ in }{\Omega},\\u=0,&&\text{ on }{\partial}{\Omega},\end{array}$$ and prove similar result for certain nonlinear term f(u).

Bond graph modeling approach for piezoelectric transducer design (압전 트랜스듀서 설계를 위한 bond graph 모델링)

  • 문원규
    • 제어로봇시스템학회:학술대회논문집
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    • 1997.10a
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    • pp.265-271
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    • 1997
  • A bond graph modeling approach which is equivalent to a finite element method is formulated in the case of the piezoelectric thickness vibrator. This formulation suggests a new definition of the generalized displacements for a continuous system as well as the piezoelectric thickness vibrator. The newly defined coordinates are illustrated to be easily interpreted physically and easily used in analysis of the system performance. Compared to the Mason equivalent circuit model, the bond graph model offers the primary advantage of physical realizability. Compared to circuit models based on standard discrete electrical elements, the main advantage of the bond graph model is a greater physical accuracy because of the use of multiport energic elements. While results are presented here for the thickness vibrator, the modeling method presented is general in scope and can be applied to arbitrary physical systems.

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