Numerical Calculation of Fractional Derivatives for the Sinc Functions via Legendre Polynomials Abbas Saadatmandi ‎Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎‎University of Kashan, ‎Kashan‎, ‎Iran author Ali Khani Department of Mathematics, Faculty of Sciences, Azarbaijan Shahid Madani University, Tabriz, Iran author Mohammad Reza Azizi Department of Mathematics, Faculty of Sciences, Azarbaijan Shahid Madani University, Tabriz, Iran author text article 2020 eng ‎This 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. Mathematics Interdisciplinary Research University of Kashan 2538-3639 5 v. 2 no. 2020 71 86 http://mir.kashanu.ac.ir/article_96936_06da74b6baa7fbe772ca4a6139f4606c.pdf dx.doi.org/10.22052/mir.2018.96632.1074 On L(d,1)-labelling of Trees Irena Hrastnik Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia author Janez Žerovnik Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia author text article 2020 eng Given a graph G and a positive integer d, an L(d,1)-labelling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u)-f(v)|>= d and if u and v are at distance two, then |f(u)-f(v)|>= 1. The L(d,1)-number of G, λd(G), is the minimum m such that there is an L(d,1)-labelling of G with f(V)⊆ {0,1,… ,m}. A tree T is of type 1 if λd(T)= Δ +d-1 and is of type 2 if λd(T)>= Δ+d. This paper provides sufficient conditions for λd(T)=Δ+d-1 generalizing the results of Wang  and Zhai, Lu, and Shu  for L(2,1)-labelling. Mathematics Interdisciplinary Research University of Kashan 2538-3639 5 v. 2 no. 2020 87 102 http://mir.kashanu.ac.ir/article_108519_7a815e176ecf64051ff8ea4e1a552bac.pdf dx.doi.org/10.22052/mir.2020.227370.1211 The Zagreb Index of Bucket Recursive Trees Ramin Kazemi Department of Statistics, Imam Khomeini International University, Qazvin, I. R. Iran author Ali Behtoei Department of Pure Mathematics, Imam Khomeini International University, Qazvin, I. R. Iran author Akram Kohansal Department of Statistics, Imam Khomeini International University, Qazvin, I. R. Iran author text article 2020 eng ‎Bucket 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. Mathematics Interdisciplinary Research University of Kashan 2538-3639 5 v. 2 no. 2020 103 111 http://mir.kashanu.ac.ir/article_109509_f3b0cb23045fab361dda93a3b444fa84.pdf dx.doi.org/10.22052/mir.2020.204312.1166