University of KashanMathematics Interdisciplinary Research2538-36395220201201Numerical Calculation of Fractional Derivatives for the Sinc Functions via Legendre Polynomials71869693610.22052/mir.2018.96632.1074ENAbbasSaadatmandiDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran0000-0002-7744-7770AliKhaniDepartment of Mathematics, Faculty of Sciences, Azarbaijan Shahid Madani University, Tabriz, IranMohammad RezaAziziDepartment of Mathematics, Faculty of Sciences, Azarbaijan Shahid Madani University, Tabriz, IranJournal Article20170827This paper provides the fractional derivatives of the Caputo type for the sinc functions. It allows to use efficient numerical method for solving fractional differential equations. At first, some properties of the sinc functions and Legendre polynomials required for our subsequent development are given. Then we use the Legendre polynomials to approximate the fractional derivatives of sinc functions. Some numerical examples are introduced to demonstrate the reliability and effectiveness of the introduced method.University of KashanMathematics Interdisciplinary Research2538-36395220201201On L(d,1)-labelling of Trees8710210851910.22052/mir.2020.227370.1211ENIrenaHrastnikFaculty of Mechanical Engineering,
University of Maribor,
Maribor, SloveniaJanezŽerovnikFaculty of Mechanical Engineering,
University of Ljubljana,
Ljubljana, SloveniaJournal Article20200417Given a graph <em>G</em> and a positive integer <em>d</em>, an L(<em>d</em>,1)-labelling of <em>G</em> is a function <em>f</em> that assigns to each vertex of <em>G</em> a non-negative integer such that if two vertices u and v are adjacent, then |<em>f</em>(<em>u</em>)-<em>f</em>(<em>v</em>)|>= <em>d</em> and if <em>u</em> and <em>v</em> are at distance two, then |<em>f</em>(<em>u</em>)-<em>f</em>(<em>v</em>)|>= 1. The L(<em>d</em>,1)-number of <em>G</em>, <em>λ<sub>d</sub></em>(<em>G</em>), is the minimum <em>m</em> such that there is an L(<em>d</em>,1)-labelling of <em>G</em> with <em>f</em>(<em>V</em>)<span>⊆</span> {0,1,… ,<em>m</em>}. A tree T is of type 1 if <em>λ<sub>d</sub></em>(<em>T</em>)= Δ +<em>d</em>-1 and is of type 2 if λ<sub>d</sub>(<em>T</em>)>= Δ+<em>d</em>. This paper provides sufficient conditions for λ<sub>d</sub>(<em>T</em>)=Δ+<em>d</em>-1 generalizing the results of Wang [11] and Zhai, Lu, and Shu [12] for L(2,1)-labelling.University of KashanMathematics Interdisciplinary Research2538-36395220201201The Zagreb Index of Bucket Recursive Trees10311110950910.22052/mir.2020.204312.1166ENRaminKazemiDepartment of Statistics, Imam Khomeini International University, Qazvin, I. R. IranAliBehtoeiDepartment of Pure Mathematics, Imam Khomeini International University, Qazvin, I. R. IranAkramKohansalDepartment of Statistics, Imam Khomeini International University, Qazvin, I. R. Iranhttps://orcid.org/0000-0002-1894-411XJournal Article20191007Bucket recursive trees are an interesting and natural generalization of recursive trees. In this model the nodes are buckets that can hold up to b>= 1 labels. The (modified) Zagreb index of a graph is defined as the sum of the squares of the outdegrees of all vertices in the graph. We give the mean and variance of this index in random bucket recursive trees. Also, two limiting results on this index are given.