University of IsfahanTransactions on Combinatorics2251-865712120230301Maximum second Zagreb index of trees with given Roman domination number1102629010.22108/toc.2022.128323.1848ENAyu Ameliatul ShahilahAhmad JamriSpecial Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Ocean Engineering Technology and
Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, MalaysiaRoslanHasniSpecial Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Ocean Engineering Technology and
Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, MalaysiaMuhammadKamran JamilDepartment of Mathematics, Riphah International University, Lahore, PakistanDoost AliMojdehDepartment of Mathematics, University of Mazandran, Babolsar, Iran0000-0001-9373-3390Journal Article20210424Chemical study regarding total $\pi$-electron energy with respect to conjugated molecules has focused on the second Zagreb index of graphs. Moreover, in the last half-century, it has gotten a lot of attention. The relationship between the Roman domination number and the second Zagreb index is investigated in this study. We characterize the trees with the maximum second Zagreb index among those with the given Roman domination number.https://toc.ui.ac.ir/article_26290_6aa308a9477fd4973448d54df8f36ff2.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712120230301Conditional probability of derangements and fixed points11262629110.22108/toc.2022.131705.1941ENSamGutmannDepartment of Mathematics, Northeastern University, 360 Huntington Ave, Boston, MA, USA.Mark D.MixerSchool of Computing and Data Science, Wentworth Institute of Technology, 550 Huntington Ave, Boston, MA, USA.StevenMorrowSchool of Computing and Data Science, Wentworth Institute of Technology, 550 Huntington Ave, Boston, MA, USA.Journal Article20211201The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper, we consider the conditional probability that the $(k+1)^{st}$ point is fixed, given there are no fixed points in the first $k$ points. We prove that when $n \neq 3$ and $k \neq 1$, this probability is a decreasing function of both $k$ and $n$. Furthermore, it is proved that this conditional probability is well approximated by $\frac{1}{n} - \frac{k}{n^2(n-1)}$. Similar results are also obtained about the more general conditional probability that the $(k+1)^{st}$ point is fixed, given that there are exactly $d$ fixed points in the first $k$ points.https://toc.ui.ac.ir/article_26291_f825454f21ba5f39cbf58e4058b7d906.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712120230301Some chemical indices related to the number of triangles27352629210.22108/toc.2022.128093.1843ENMihrigulWaliSchool of Mathematical Sciences, Xiamen University, 361005, Fuzhou, Xiamen, P. R. China.381883720@qq.comJournal Article20210410Many chemical indices have been invented in theoretical chemistry, such as the Zagreb index, the Lanzhou index, the forgotten index, the Estrada index etc. In this paper, we show that the first Zagreb index is only related to the sum of the number of triangles in a graph and the number of triangles in its complement. Moreover, we determine the sum of the first and second Zagreb index, the Lanzhou index and the forgotten index for a graph and its complement in terms of the number of triangles in a graph and the number of triangles in its complement. Finally, we estimate the Estrada index in terms of order, size, maximum degree and the number of triangles.https://toc.ui.ac.ir/article_26292_c95daa4343854df9a61b2113d46444ca.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712120230301The Mostar and Wiener index of alternate Lucas cubes37462629310.22108/toc.2022.130675.1912ENÖmerEğecioğluDepartment of Computer Science, University of California Santa Barbara, CA 93106, USA0000-0002-6070-761XElifSaygDepartment of Mathematics and Science Education, Hacettepe University, 06800, Ankara, Turkey0000-0001-8811-4747ZülfükarSaygiDepartment of Mathematics, TOBB University of Economics and Technology, 06560, Ankara, Turkey0000-0002-7575-3272Journal Article20210921The Wiener index and the Mostar index quantify two distance related properties of connected graphs: the Wiener index is the sum of the distances over all pairs of vertices and the Mostar index is a measure of how far the graph is from being distance-balanced. These two measures have been considered for a number of interesting families of graphs. In this paper, we determine the Wiener index and the Mostar index of alternate Lucas cubes. Alternate Lucas cubes form a family of interconnection networks whose recursive construction mimics the construction of the well-known Fibonacci cubes.https://toc.ui.ac.ir/article_26293_10c85bae888b9e639d13806fc13d60b0.pdfUniversity of IsfahanTransactions on Combinatorics2251-865712120230301Generalized barred preferential arrangements47632633710.22108/toc.2022.130037.1894ENJoséA. AdellDepartamento de Métodos Estadı́sticos, Facultad de Ciencias, Universidad Zaragoza, C. de Pedro Cerbuna, 12, 50009,
Zaragoza, SpainBeátaBényiDepartment of Hydraulic Engineering, Faculty of Water Sciences, H-6500, Bajcsy-Zsilinszky utca 12–14, University of Public Service Baja, HungaryVenkatMuraliDepartment of Mathematics, Rhodes University, Grahamstown, 6139, South AfricaSithembeleNkonkobeDepartment of Mathematical Sciences, Sol Plaatje University, Kimberley, 8301, South AfricaJournal Article20210812We investigate a generalization of Fubini numbers. We present the combinatorial interpretation as barred preferential arrangements with some additional conditions on the blocks. We provide a proof for a generalization of Nelsen's Theorem. We consider these numbers from a probabilistic view point and demonstrate how they can be written in terms of the expectation of random descending factorial involving the negative binomial process.https://toc.ui.ac.ir/article_26337_53dca73077c31cf7d02921c47a1f0b5d.pdf