Proyecciones (Antofagasta, On line) <p align="justify">La revista&nbsp;Proyecciones. Journal of Mathematics es una publicación científica, sin fines de lucro, oficial de la Universidad Católica del Norte, Antofagasta, Chile. Fue fundada en 1982 y depende del Departamento de Matemáticas de la Universidad Católica del Norte.<br>Proyecciones. Journal of Mathematics edita un volumen con 5 números al año.</p> Universidad Católica del Norte. en-US Proyecciones (Antofagasta, On line) 0717-6279 Special Issue on Mathematical Computation in Combinatorics and Graph Theory Mohammad Reza Farahani Copyright (c) 2020 Mohammad Reza Farahani 2020-07-28 2020-07-28 39 4 i i Rainbow and strong rainbow connection number for some families of graphs <p><em>Let G be a nontrivial connected graph. Then G is called a rainbow connected graph if there exists a coloring c : E(G) ? {1, 2, ..., k}, k ? N, of the edges of G, such that there is a u ? v rainbow path between every two vertices of G, where a path P in G is a rainbow path if no two edges of P are colored the same. The minimum k for which there exists such a k-edge coloring is the rainbow connection number rc(G) of G. If for every pair u, v of distinct vertices, G contains a rainbow u ? v geodesic, then G is called strong rainbow connected. The minimum k for which G is strong rainbow-connected is called the strong rainbow connection number src(G) of G.</em></p> <p><em>The exact rc and src of the rotationally symmetric graphs are determined.</em></p> Yaqoub Ahmed Khan Muhammad Naeem Muhammad Kamran Siddiqui Mohammad Reza Farahani Copyright (c) 2020 Yaqoub Ahmed Khan, Muhammad Naeem, Muhammad Kamran Siddiqui, Mohammad Reza Farahani 2020-07-28 2020-07-28 39 4 737 747 10.22199/issn.0717-6279-2020-04-0046 M-Polynomial and topological indices of benzene ring embedded in p-type surface network <p><em>The representation of chemical compounds and chemical networks with the M-polynomials is a new idea and it gives nice and good results of the topological indices. These results are used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities.</em></p> <p><em>Particular attention is paid to the derivation of the M polynomia- for the benzene ring embedded in the P-type surface network in 2D. Furthermore, the topological indices based on the degrees are also derived by using the general form of M-polynomial of benzene ring embedded in the P-type surface network BR(m, n). In the end, the graphical representation and comparison of the M-polynomial and the topological indices of benzene ring embedded in P-type surface network in 2D are described.</em></p> Murat Cancan Suleyman Ediz Abdul Baig Waqas Khalid Copyright (c) 2020 Murat Cancan, Suleyman Ediz, Abdul Baig, Waqas Khalid 2020-07-28 2020-07-28 39 4 749 768 10.22199/issn.0717-6279-2020-04-0047 Molecular descriptors of certain OTIS interconnection networks <p><em>Network theory as an important role in the field of electronic and electrical engineering, for example, in signal processing, networking, communication theory, etc. The branch of mathematics known as Graph theory found remarkable applications in this area of study. A topological index (TI) is a real number attached with graph networks and correlates the chemical networks with many physical and chemical properties and chemical reactivity. The Optical Transpose Interconnection System (OTIS) network has received considerable attention in recent years and has a special place among real world architectures for parallel and distributed systems. In this report, we compute redefined first, second and third Zagreb indices of OTIS swapped and OTIS biswapped networks. We also compute some Zagreb polynomials of understudy Networks.</em></p> Murat Cancan Iftikhar Ahmad Sarfarz Ahmad Copyright (c) 2020 Murat Cancan, Iftikhar Ahmad, Sarfarz Ahmad 2020-07-28 2020-07-28 39 4 769 786 10.22199/issn.0717-6279-2020-04-0048 Edge irregularity strength of certain families of comb graph <p><em>Edge irregular mapping or vertex mapping h : V (U ) ?? {1, 2, 3, 4, ..., s} is a mapping of vertices in such a way that all edges have distinct weights. We evaluate weight of any edge by using equation wt<sub>h</sub>(cd) = h(c)+h(d), ?c, d ? V (U ) and ?cd ? E(U ). Edge irregularity strength denoted by es(U ) is a minimum positive integer use to label vertices to form edge irregular labeling. In this paper, we find exact value of edge irregularity strength of different families of comb graph.</em></p> Xiujun Zhang Murat Cancan Muhammad Faisal Nadeem Muhammad Imran Copyright (c) 2020 Xiujun Zhang, Murat Cancan, Muhammad Faisal Nadeem, Muhammad Imran 2020-07-28 2020-07-28 39 4 787 797 10.22199/issn.0717-6279-2020-04-0049 Bounds for neighborhood Zagreb index and its explicit expressions under some graph operations <p><em>Topological indices are useful in QSAR/QSPR studies for modeling biological and physiochemical properties of molecules. The neighborhood Zagreb index (M<sub>N</sub>) is a novel topological index having good correlations with some physiochemical properties. For a simple connected graph G, the neighborhood Zagreb index is the totality of square of ?<sub>G</sub>(v) over the vertex set, where ?<sub>G</sub>(v) is the total count of degrees of all neighbors of v in G. In this report, some bounds are established for the neighborhood Zagreb index. Some explicit expressions of the index for some graph operations are also computed, which are used to obtain the index for some chemically significant molecular graphs. </em></p> Sourav Mondal Muhammad Arfan Ali Nilanjan De Anita Pal Copyright (c) 2020 Sourav Mondal, Muhammad Arfan Ali , Nilanjan De , Anita Pal 2020-07-28 2020-07-28 39 4 799 819 10.22199/issn.0717-6279-2020-04-0050 The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane <p><em>A description of Endomorphisms of the translation group is introduced in an affine plane, will define the addition and composition of the set of endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving endomorphism algebra in affine plane, and prove that the set of Trace-preserving endomorphism associated with the ’addition’ action forms a commutative group. We also try to prove that the set of trace-preserving endomorphism, together with the two actions, in it, ’addition’ and ’composition’ forms an associative and unitary ring.</em></p> Orgest Zaka Mohanad. A. Mohammed Copyright (c) 2020 Orgest Zaka, Mohanad. A. Mohammed 2020-07-28 2020-07-28 39 4 821 834 10.22199/issn.0717-6279-2020-04-0051 Skew-field of trace-preserving endomorphisms, of translation group in affine plane <p><em>We will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving endomorphism, with the help of which we can construct the inverse trace-preserving endomorphisms of every trace-preserving endomorphism. At the end of this paper we have proven that the set of tracepreserving endomorphisms together with the actions of ’addition’ and ’composition’ (which is in the role of ’multiplication’) forms a skewfield.</em></p> Orgest Zaka Mohanad A. Mohammed Copyright (c) 2020 Orgest Zaka, Mohanad A. Mohammed 2020-07-28 2020-07-28 39 4 823 850 10.22199/issn.0717-6279-2020-04-0052 Redefined Zagreb indices of Rhombic, triangular, Hourglass and Jagged-rectangle benzenoid systems <p><em>In the fields of mathematical chemistry and chemical graph theory, a topological index generally called a connectivity index is a kind of a molecular descriptor that is calculated in perspective of the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which depict its topology and are graph invariant up to graph isomorphism. Topological indices are used for example in the progression of quantitative structure-activity relationships (QSARs) in which the common activity or distinctive properties of atoms are connected with their molecular structure. There are in excess of 140 topological indices but none of them totally describe the molecular compound completely so there is dependably a space to characterize and register new topological indices. Benzenoid Systems are utilized basically as an intermediate to make different synthetic compounds. In this report we aim to compute redefined Zagreb indices for Zigzag, Rhombic, triangular, Hourglass and Jagged-rectangle Benzenoid systems.</em></p> Mohanad A. Mohammed Raad Haoer Ashaq Ali Maqbool Ahmad Mohammad Reza Farahani Saima Nazeer Copyright (c) 2020 Mohanad A. Mohammed, Raad Haoer, Ashaq Ali, Maqbool Ahmad, Mohammad Reza Farahani, Saima Nazeer 2020-07-28 2020-07-28 39 4 851 867 10.22199/issn.0717-6279-2020-04-0053 Distance and eccentricity based polynomials and indices of m-level Wheel graph <p><em>Distance and degree based topological polynomial and indices of molecular graphs have various applications in chemistry, computer networking and pharmacy. In this paper, we give hosoya polynomial, Harary polynomial, Schultz polynomial, modified Schultz polynomial, eccentric connectivity polynomial, modified Wiener index, modified hyper Wiener index, generalized Harary index, multiplicative Wiener index, Schultz index, modified Schultz index, eccentric connectivity index of generalized wheel networks Wn,m. We also give pictorial representation of computed topological polynomials and indices on the involved parameters m and n.</em></p> Murat Cancan Muhammad Hussain Haseeb Ahmad Copyright (c) 2020 Murat Cancan, Muhammad Hussain, Haseeb Ahmad 2020-07-28 2020-07-28 39 4 869 885 10.22199/issn.0717-6279-2020-04-0054 Study of topology of block shift networks via topological indices <p><em>Topological indices(TIs) are important numerical number associate with the molecular graph of a chemical structure/compound because due to these parameters, one can guess almost all properties of concerned structure/compound with our performing experiments. In recent years, huge amount work has been done for calculating degreedependent indices for different structures/compouds. In order to compute TIs, one need to do many calculations. Our aim of this paper is to present a simple method to compute degree-dependent TIs. We computed M-polynomials for Block Shift Networks and with the help of this simple algebraic polynomials, we recovered nine important TIs for Block Shift Networks. Our work is important for chemists, physicians and pharmaceutical industry.</em></p> Murat Cancan Iftikhar Ahmad Sarfarz Ahmad Copyright (c) 2020 Murat Cancan, Iftikhar Ahmad, Sarfarz Ahmad 2020-07-28 2020-07-28 39 4 887 902 10.22199/issn.0717-6279-2020-04-0055 Irregularity indices for line graph of Dutch windmill graph <p><em>Among topological descriptors topological indices are significant and they have a conspicuous role in chemistry. Dutch Windmill graph D<sup>x</sup><sub>y</sub> can be obtain by taking x copies of cycle C<sub>y</sub> with a vertex in common. In this paper, we will compute some irregularity índices that are useful in quantitative structure activity relationship for Line Graph of Dutch Windmill graph.</em></p> Mohanad A. Mohammed Suad Younus A. AL-Mayyahi AL-Mayyahi Abaid ur Rehman Virk Hafiz Mutee ur Rehman Copyright (c) 2020 Mohanad A. Mohammed, Suad Younus A. AL-Mayyahi AL-Mayyahi, Abaid ur Rehman Virk, Hafiz Mutee ur Rehman 2020-07-28 2020-07-28 39 4 903 918 10.22199/issn.0717-6279-2020-04-0056 Some resistance distance and distance-based graph invariants and number of spanning trees in the tensor product of P2 and Kn <p><em>The resistance distance (Kirchhoff index and multiplicative Kirchhoff index) and distance-based (Wiener index and Gutman index) graph invariants of ?<sub>n</sub> = P<sub>2</sub> ×K<sub>n</sub> are considered. Firstly by using the decomposition theorem, we procure the Laplacian and Normalized Laplacian spectrum for graph ?<sub>n</sub>, respectively. Based on which, we can procured the formulae for the number of spanning trees and some resistance distance and distance-based graph invariants of graph ?<sub>n</sub>. Also, it is very interesting to see that when n tends to infinity, Kf (?<sub>n</sub>) is a polynomial and W (?<sub>n</sub>) is a quadratic polynomial.</em></p> Muhammad Shoaib Sardar Murat Cancan Süleyman Ediz Wasim Sajjad Copyright (c) 2020 Muhammad Shoaib Sardar, Murat Cancan, Süleyman Ediz, Wasim Sajjad 2020-07-28 2020-07-28 39 4 919 932 10.22199/issn.0717-6279-2020-04-0057 Programming with MATLAB to color latin squares <p><em>With a Matlab programming we will find the chromatic number for all Latin squares of order smaller than 7. Previously, a manual algorithm for coloring the Latin square was provided. This algoritm determined the chromatic number of some special classes of Latin squares such as Cyclic or Dihedral, so, we tried to speed up the process of this algorithm with a programming.</em></p> A. Shokri M. Golriz M. Alaeiyan Copyright (c) 2020 A. Shokri, M. Golriz, M. Alaeiyan 2020-07-28 2020-07-28 39 4 933 943 10.22199/issn.0717-6279-2020-04-0058 Modules whose partial endomorphisms have a ?-small kernels <p><em>Let R be a commutative ring and M a unital R-module. A submodule N is said to be ?-small, if whenever N + L = M with M/L is singular, we have L = M. M is called ?-small monoform if any of its partial endomorphism has ?-small kernel. In this paper, we introduce the concept of ?-small monoform modules as a generalization of monoform modules and give some of their properties, examples and characterizations.</em></p> Papa Cheikhou Diop Abdoul Djibril Diallo Copyright (c) 2020 Papa Cheikhou Diop, Abdoul Djibril Diallo 2020-07-28 2020-07-28 39 4 945 962 10.22199/issn.0717-6279-2020-04-0059 Type IV codes over a non-local non-unital ring <p><em>There is a local ring H of order 4, without identity for the multiplication, defined by generators and relations as</em></p> <p><em>H =</em><em>?</em><em>a, b | 2a = 2b = 0, a<sup>2</sup> = 0, b<sup>2</sup> = b, ab = ba = 0</em><em>?</em><em>.</em></p> <p><em>We classify self orthogonal codes of length n and size 2<sup>n</sup> (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify quasi Type IV codes (a subclass of Type IV codes, viz QSD codes with even weights) up to n = 6. </em></p> Adel Alahmadi Amani Alkathiry Alaa Altassan Widyan Basaffar Alexis Bonnecaze Hatoon Shoaib Patrick Solé Copyright (c) 2020 Adel Alahmadi, Amani Alkathiry, Alaa Altassan, Widyan Basaffar, Alexis Bonnecaze, Hatoon Shoaib, Patrick Solé 2020-07-28 2020-07-28 39 4 963 978 10.22199/issn.0717-6279-2020-04-0060 Topological properties of four types of porphyrin dendrimers <p><em>A chemical compound can be represented as a chemical graph. A topological index of a (chemical) graph is a numeric value of a graph which characterize its topology and is usually graph invariant. The Zagreb indices, Randi? index and sum-connectivity indices are useful in the study of anti-inflammatory activities, boiling point, molecular complexity heterosystems of certain chemical instances, and in elsewhere. In this paper, we calculate the mentioned topological indices of some infinite classes of prophyrin dendrimers. </em></p> Abdul Jalil M. Khalaf Aisha Javed Muhammad Kamran Jamil Mehdi Alaeiyan Mohammad Reza Farahani Copyright (c) 2020 Abdul Jalil M. Khalaf, Aisha Javed, Muhammad Kamran Jamil, Mehdi Alaeiyan, Mohammad Reza Farahani 2020-07-28 2020-07-28 39 4 979 993 10.22199/issn.0717-6279-2020-04-0061 Super antimagic total labeling under duplication operations <p><em>For a graph G the duplication operation of a vertex v by a new edge e = uw results in a new graph G’ such that N (u) = {v, w} and N (w) = {v, u}. The duplication operation of an edge e = uv by a new vertex w results in a new graph G’’ such that N (w) = {u, v}. In this article we have discussed that the properties of a graph, with minimum degree 2 of any vertex, to be super vertex-antimagic total and to be super edge-antimagic total are invariant under the above duplication operations. Also, we have discussed on the existence of the so-called k super vertex-antimagic total vertex modifications and k super edge-antimagic total edge modifications for graphs. </em></p> Abdul Jalil M. Khalaf Muhammad Naeem Muhammad Kamran Siddiqui Abdul Qudair Baig Copyright (c) 2020 Muhammad Naeem, Muhammad Kamran Siddiqui, Abdul Qudair Baig 2020-07-28 2020-07-28 39 4 995 1003 10.22199/issn.0717-6279-2020-04-0062 The lower bound and exact value of the information rate of some developed graph access structures <p><em>Various studies have focused on secret sharing schemes and in all of them, each shareholder is interested in a shorter share. The information rate of a secret sharing scheme shows the ratio between the size of the secret to the maximum number of shares given to each shareholder. In this regard, the researchers investigated the optimal information rate of the graph access structure. This paper aims to discover the exact values for the optimal information rates of the two graph access structures, which remained as open problems in Van Dijk’s paper. Furthermore, we introduced the developed multipartite graph and the developed cycle graph and calculated the exact value of their optimal information rate. Moreover, we presented a lower bound on the information rate for other developed graph access structures. </em></p> Abbas Cheraghi Copyright (c) 2020 Abbas Cheraghi 2020-07-28 2020-07-28 39 4 1005 1017 10.22199/issn.0717-6279-2020-04-0063 Entire forgotten topological index of graphs <p><em>The Forgotten topological index or F-index is defined as the sum of cubes of the degrees of vertices of a graph. For this classical F-index, we ignore the intermolecular forces that exist between the atoms and bonds and consider only the intermolecular forces between the atoms of a molecule. In this paper, we introduce a new graph invariant called “Entire Forgotten topological index” or “Entire F-index”, that includes incidency of edges and vertices in addition to the adjacency of the vertices. We obtain some important properties of the index and also establish formulae of this newly defined index for some operations of graphs.</em></p> A. Bharali Amitav Doley Jibonjyoti Buragohain Copyright (c) 2020 A. Bharali, Amitav Doley, Jibonjyoti Buragohain 2020-07-28 2020-07-28 39 4 1019 1032 10.22199/issn.0717-6279-2020-04-0064 Edge even graceful labeling of torus grid graph <p><em>We study the family of torus grid graphs. We also obtain necessary and sufficent conditions to be edge even graceful labeling for all of the cases of every member of this family.</em></p> Salama Nagy Daoud Wedad Saleh Copyright (c) 2020 Salama Nagy Daoud, Wedad Saleh 2020-07-28 2020-07-28 39 4 1033 1082 10.22199/issn.0717-6279-2020-04-0065 Quasi self-dual codes over non-unital rings of order six <p><em>There exist two semi-local rings of order 6 without identity for the multiplication. We classify up to coordinate permutation self-orthogonal codes of length n and size 6<sup>n/2 </sup>over these rings (called here quasi self-dual codes or QSD) till the length n = 8. To any such code is attached canonically a </em><em>?</em><em><sub>6</sub></em><em>-code, which, when self-dual, produces an unimodular lattice by Construction A. </em></p> Adel Alahmadi Amani Alkathiry Alaa Altassan Widyan Basaffar Alexis Bonnecaze Hatoon Shoaib Patrick Solé Copyright (c) 2020 Adel Alahmadi, Amani Alkathiry, Alaa Altassan, Widyan Basaffar, Alexis Bonnecaze, Hatoon Shoaib, Patrick Solé 2020-07-28 2020-07-28 39 4 1083 1095 10.22199/issn.0717-6279-2020-04-0066 On additive maps of MA-semirings with involution <p><em>We extend the concept of *-derivations of rings to a certain class of semirings called MA-semirings and establish some results on commutativity forced by the *-derivations satisfying different criteria. We specially focus on the results on certain conditions under which additive mappings become Jordan *-derivations. </em></p> Liaqat Ali Muhammad Aslam Yaqoub Ahmed Khan Copyright (c) 2020 Liaqat Ali, Muhammad Aslam, Yaqoub Ahmed Khan 2020-07-28 2020-07-28 39 4 1097 1112 10.22199/issn.0717-6279-2020-04-0067