Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.4.561
S. Cichacz, Agnieszka G�rlich, Andrea Semani�ov�-Fe�ov��kov�
For a graph (G) its distance vertex irregularity strength is the smallest integer (k) for which one can find a labeling (f: V(G)to {1, 2, dots, k}) such that [ sum_{xin N(v)}f(x)neq sum_{xin N(u)}f(x)] for all vertices (u,v) of (G), where (N(v)) is the open neighborhood of (v). In this paper we present some upper bounds on distance vertex irregularity strength of general graphs. Moreover, we give upper bounds on distance vertex irregularity strength of hypercubes and trees.
{"title":"Upper bounds on distance vertex irregularity strength of some families of graphs","authors":"S. Cichacz, Agnieszka G�rlich, Andrea Semani�ov�-Fe�ov��kov�","doi":"10.7494/opmath.2022.42.4.561","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.4.561","url":null,"abstract":"For a graph (G) its distance vertex irregularity strength is the smallest integer (k) for which one can find a labeling (f: V(G)to {1, 2, dots, k}) such that [ sum_{xin N(v)}f(x)neq sum_{xin N(u)}f(x)] for all vertices (u,v) of (G), where (N(v)) is the open neighborhood of (v). In this paper we present some upper bounds on distance vertex irregularity strength of general graphs. Moreover, we give upper bounds on distance vertex irregularity strength of hypercubes and trees.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.6.867
Y. Shoukaku
{"title":"Forced oscillation and asymptotic behavior of solutions of linear differential equations of second order","authors":"Y. Shoukaku","doi":"10.7494/opmath.2022.42.6.867","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.6.867","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.1.93
L. Saha, Mithun Basak, Kalishankar Tiwary
Summary: For a simple connected graph G = ( V, E ) and an ordered subset W = { w 1 , w 2 , . . . , w k } of V , the code of a vertex v ∈ V , denoted by code( v ) , with respect to W is a k -tuple ( d ( v, w 1 ) , . . . , d ( v, w k )) , where d ( v, w t ) represents the distance between v and w t . The set W is called a resolving set of G if code( u ) ̸ = code( v ) for every pair of distinct vertices u and v . A metric basis of G is a resolving set with the minimum cardinality. The metric dimension of G is the cardinality of a metric basis and is denoted by β ( G ) . A set F ⊂ V is called fault-tolerant resolving set of G if F { v } is a resolving set of G for every v ∈ F . The fault-tolerant metric dimension of G is the cardinality of a minimal fault-tolerant resolving set. In this article, a complete characterization of metric bases for G 2 mn has been given. In addition, we prove that the fault-tolerant metric dimension of G 2 mn is 4 if m + n is even. We also show that the fault-tolerant metric dimension of G 2 mn is at least 5 and at most 6 when m + n is
摘要:对于简单连通图G = (V, E)和有序子集W = {w1, w2，…， w k} (V)，顶点V∈V的码，记作code(V)，关于w是一个k元组(d (V, w 1)，…， d (v, w k))，其中d (v, w t)表示v和w t之间的距离。如果对每一对不同的顶点u和v都有code(u) = code(v)，则集合W称为G的解析集。G的度量基是具有最小基数的解析集。G的度量维数是度量基的基数，用β (G)表示。如果F {V}是G对每一个V∈F的解析集，则集合F∧V称为G的容错解析集。G的容错度量维是最小容错解析集的基数。本文给出了g2mn的度量基的完整表征。此外，我们还证明了当m + n为偶数时，g2mn的容错度量维数为4。我们还证明了当m + n为时，g2mn的容错度量维数最小为5，最大为6
{"title":"All metric bases and fault-tolerant metric dimension for square of grid","authors":"L. Saha, Mithun Basak, Kalishankar Tiwary","doi":"10.7494/opmath.2022.42.1.93","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.1.93","url":null,"abstract":"Summary: For a simple connected graph G = ( V, E ) and an ordered subset W = { w 1 , w 2 , . . . , w k } of V , the code of a vertex v ∈ V , denoted by code( v ) , with respect to W is a k -tuple ( d ( v, w 1 ) , . . . , d ( v, w k )) , where d ( v, w t ) represents the distance between v and w t . The set W is called a resolving set of G if code( u ) ̸ = code( v ) for every pair of distinct vertices u and v . A metric basis of G is a resolving set with the minimum cardinality. The metric dimension of G is the cardinality of a metric basis and is denoted by β ( G ) . A set F ⊂ V is called fault-tolerant resolving set of G if F { v } is a resolving set of G for every v ∈ F . The fault-tolerant metric dimension of G is the cardinality of a minimal fault-tolerant resolving set. In this article, a complete characterization of metric bases for G 2 mn has been given. In addition, we prove that the fault-tolerant metric dimension of G 2 mn is 4 if m + n is even. We also show that the fault-tolerant metric dimension of G 2 mn is at least 5 and at most 6 when m + n is","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.2.179
Guenbo Hwang, Byungsoo Moon
We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural ((2+1))-dimensional generalization of the ((1+1))-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the (d)-bar formalism, also known as Pompieu's formula. In addition, we characterize the (d)-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.
{"title":"The d-bar formalism for the modified Veselov-Novikov equation on the half-plane","authors":"Guenbo Hwang, Byungsoo Moon","doi":"10.7494/opmath.2022.42.2.179","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.2.179","url":null,"abstract":"We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural ((2+1))-dimensional generalization of the ((1+1))-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the (d)-bar formalism, also known as Pompieu's formula. In addition, we characterize the (d)-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.1.5
Messaouda Ben Attia, E. Zaouche, M. Bousselsal
By choosing convenient test functions and using the method of doubling variables, we prove the uniqueness of the solution to a nonlinear evolution dam problem in an arbitrary heterogeneous porous medium of (mathbb{R}^n) ((nin {2,3})) with an impermeable horizontal bottom.
{"title":"Uniqueness of solution of a nonlinear evolution dam problem in a heterogeneous porous medium","authors":"Messaouda Ben Attia, E. Zaouche, M. Bousselsal","doi":"10.7494/opmath.2022.42.1.5","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.1.5","url":null,"abstract":"By choosing convenient test functions and using the method of doubling variables, we prove the uniqueness of the solution to a nonlinear evolution dam problem in an arbitrary heterogeneous porous medium of (mathbb{R}^n) ((nin {2,3})) with an impermeable horizontal bottom.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.1.55
N. Indrajith, J. Graef, E. Thandapani
The authors present Kneser-type oscillation criteria for a class of advanced type second-order difference equations. The results obtained are new and they improve and complement known results in the literature. Two examples are provided to illustrate the importance of the main results.
{"title":"Kneser-type oscillation criteria for second-order half-linear advanced difference equations","authors":"N. Indrajith, J. Graef, E. Thandapani","doi":"10.7494/opmath.2022.42.1.55","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.1.55","url":null,"abstract":"The authors present Kneser-type oscillation criteria for a class of advanced type second-order difference equations. The results obtained are new and they improve and complement known results in the literature. Two examples are provided to illustrate the importance of the main results.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.3.393
E. Attia, B. El-Matary
We study the oscillation of first-order linear difference equations with non-monotone deviating arguments. Iterative oscillation criteria are obtained which essentially improve, extend, and simplify some known conditions. These results will be applied to some numerical examples.
{"title":"New aspects for the oscillation of first-order difference equations with deviating arguments","authors":"E. Attia, B. El-Matary","doi":"10.7494/opmath.2022.42.3.393","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.3.393","url":null,"abstract":"We study the oscillation of first-order linear difference equations with non-monotone deviating arguments. Iterative oscillation criteria are obtained which essentially improve, extend, and simplify some known conditions. These results will be applied to some numerical examples.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.3.415
S. Hamouda, S. Mahmoudi
This paper is devoted to the study of the growth of solutions of certain class of linear fractional differential equations with polynomial coefficients involving the Caputo fractional derivatives by using the generalized Wiman-Valiron theorem in the fractional calculus.
{"title":"Growth of solutions of a class of linear fractional differential equations with polynomial coefficients","authors":"S. Hamouda, S. Mahmoudi","doi":"10.7494/opmath.2022.42.3.415","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.3.415","url":null,"abstract":"This paper is devoted to the study of the growth of solutions of certain class of linear fractional differential equations with polynomial coefficients involving the Caputo fractional derivatives by using the generalized Wiman-Valiron theorem in the fractional calculus.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.4.573
T. Haynes, Michael A. Henning
A set (S) of vertices in an isolate-free graph (G) is a total dominating set if every vertex in (G) is adjacent to a vertex in (S). A total dominating set of (G) is minimal if it contains no total dominating set of (G) as a proper subset. The upper total domination number (Gamma_t(G)) of (G) is the maximum cardinality of a minimal total dominating set in (G). We establish Nordhaus-Gaddum bounds involving the upper total domination numbers of a graph (G) and its complement (overline{G}). We prove that if (G) is a graph of order (n) such that both (G) and (overline{G}) are isolate-free, then (Gamma_t(G) + Gamma_t(overline{G}) leq n + 2) and (Gamma_t(G)Gamma_t(overline{G}) leq frac{1}{4}(n+2)^2), and these bounds are tight.
如果(G)中的每个顶点与(S)中的一个顶点相邻，那么无隔离图(G)中的顶点集(S)就是一个总支配集。如果不包含(G)作为适当子集的总支配集，则(G)的总支配集是最小的。(G)的上总支配数(Gamma_t(G))是(G)中最小总支配集的最大基数。我们建立了涉及图(G)及其补(overline{G})的上总控制数的诺德豪斯-加德姆界。我们证明了如果(G)是一个阶为(n)的图，使得(G)和(overline{G})都是无隔离的，那么(Gamma_t(G) + Gamma_t(overline{G}) leq n + 2)和(Gamma_t(G)Gamma_t(overline{G}) leq frac{1}{4}(n+2)^2)，并且这些界是紧的。
{"title":"Nordhaus-Gaddum bounds for upper total domination","authors":"T. Haynes, Michael A. Henning","doi":"10.7494/opmath.2022.42.4.573","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.4.573","url":null,"abstract":"A set (S) of vertices in an isolate-free graph (G) is a total dominating set if every vertex in (G) is adjacent to a vertex in (S). A total dominating set of (G) is minimal if it contains no total dominating set of (G) as a proper subset. The upper total domination number (Gamma_t(G)) of (G) is the maximum cardinality of a minimal total dominating set in (G). We establish Nordhaus-Gaddum bounds involving the upper total domination numbers of a graph (G) and its complement (overline{G}). We prove that if (G) is a graph of order (n) such that both (G) and (overline{G}) are isolate-free, then (Gamma_t(G) + Gamma_t(overline{G}) leq n + 2) and (Gamma_t(G)Gamma_t(overline{G}) leq frac{1}{4}(n+2)^2), and these bounds are tight.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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