https://www.revistaproyecciones.cl/index.php/proyecciones/issue/feedProyecciones (Antofagasta, On line)2024-01-21T15:21:03+00:00Ricardo Soto Monterorsoto@ucn.clOpen Journal Systems<p align="justify">La revista 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>https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4894Dr Even vertex odd mean labeling of uniform theta graphs2021-05-16T19:28:33+00:00mohamed basher basherm_e_basher@yahoo.com<p>Let $G$ be a graph with $p$ vertices and $q$ edges. A total graph labeling $ f:V(G)\bigcup E(G)\rightarrow \{0,1,2,3,...,2q\}$ is called even vertex odd mean labeling of a graph $G$ if the vertices of the graph $G$ label by distinct even integers from the set $\{0,2,...,2q\}$ and the labels of the edges are defined as the mean of the labels of its end vertices and these labels are $2q-1$ distinct odd integers from the set $\{1,3,5,...,2q-1\}$. In this paper we investigate the even vertex odd mean labeling of uniform theta graphs.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 mohamed basher basherhttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5880Group vertex magic labeling of some special graphs2023-10-13T12:33:02+00:00mohamed basher basherm_e_basher@yahoo.com<p>For any additive abelian group $A$. Let $\mu$ be an element of $A$, a graph $G=(V,E)$ is said to be $A$-vertex magic graph if there exist a labeling function $f:V(G)\rightarrow A\setminus\{0\}$ such that $\omega(v)=\sum_{u\in N(v)} f(u)=\mu$ for any vertex $v$ of $G$, where $N(v)$ is the set of the open neighborhood of $v$. In this paper, we prove that the graphs such as wheel, Corona $C_{n}\odot mk$, subdivision of ladder and $t$-fold wheel for $t\neq n$ nor $n-2$ are $A$-vertex magic graphs. Also we prove that the subdivide wheel, helm and closed helm are $Z_{k}$-vertex magic graphs. However we prove that the triangular book and $t$-fold wheel for $t=n,n-2$ are group vertex magic graphs.<br><br></p>2024-03-11T00:00:00+00:00Copyright (c) 2024 mohamed basher basherhttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5804Some inequalities between degree- and distance-based topological indices2023-05-15T14:22:15+00:00Dr. Imran Nadeem Imranimran7355@gmail.com2024-03-11T00:00:00+00:00Copyright (c) 2024 Dr. Imran Nadeem Imranhttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5584COMPUTATION OF WIENER POLYNOMIAL AND INDEX OF LINE SUBDIVISION FRIENDSHIP AND LINE SUBDIVISION BIFRIENDSHIP GRAPHS USING MATLAB PROGRAM 2023-07-21T16:13:56+00:00Mohammed Al-Sharafialsharafi205010@gmail.comAbdu Alameri a.alameri2222@gmail.com<p>A topological index is a branch of chemical graph theory that is vital to analyzing the physio-chemical characteristics of chemical compound structures divided into a degree-based molecular structure such as Zagreb indices, a distance-based molecular structure such as Wiener index, and a mixed such as Gutman index. In this paper, some definitions, results, and examples of Wiener polynomial and index for subdivision graph of friendship, bifriendship graphs, line subdivision graph of friendship, and bifriendship graphs were introduced. Moreover, we used the MATLAB program to calculate the Wiener polynomial and index of these graphs and refer to some applications.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 Mohammed Al-Sharafi, Prof. Abdu Alameri https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/6117Tight Bounds for the $N_2$-Chromatic Number of Graphs2023-08-14T17:41:35+00:00Ian June Garcesijlgarces@ateneo.edu<p>Let $G$ be a connected graph. A vertex coloring of $G$ is an $N_2$-vertex coloring if, for every vertex $v$, the number of different colors assigned to the vertices adjacent to $v$ is at most two. The $N_2$-chromatic number of $G$ is the maximum number of colors that can be used in an $N_2$-vertex coloring of $G$. In this paper, we establish tight bounds for the $N_2$-chromatic number of a graph in terms of its maximum degree and its diameter, and characterize those graphs that attain these bounds.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 Ian June Garceshttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5401Extreme Outer Connected Geodesic Graphs2023-02-18T15:31:54+00:00Ganesamoorthy K.kvgm_2005@yahoo.co.inJayanthi Ddjayanthimahesh@gmail.com<p>For a connected graph G of order at least two, a set S of vertices in a graph G is said to be an outer connected geodetic set if S is a<br>geodetic set of G and either S = V or the subgraph induced by V − S is connected. The minimum cardinality of an outer connected geodetic set of G is the outer connected geodetic number of G and is denoted by goc(G). The number of extreme vertices in G is its extreme order ex(G). A graph G is said to be an extreme outer connected geodesic graph if goc(G) = ex(G). It is shown that for every pair a, b of integers with 0 ≤ a ≤ b and b ≥ 2, there exists a connected graph G with ex(G) = a and goc(G) = b. Also, it is shown that for positive integers r, d and k ≥ 2 with r < d ≤ 2r, there exists an extreme outer connected geodesic graph G of radius r, diameter d and outer connected geodetic number k.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 Ganesamoorthy K., Jayanthi Dhttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5984Optimal modeling of nonlinear systems: Method of variable injections2023-04-22T15:42:37+00:00JUAN PABLO SOTO QUIROSjpablocr13@gmail.comAnatoli Torokhtianatoli.torokhti@unisa.edu.au<p>Our work addresses a development and justification of the new approach to the modeling of nonlinear systems. Let $\f$ be an unknown input-output map of the system with a random input and output $\y$ and $\x$, respectively. It is assumed that $\y$ and $\x$ are available and covariance matrices formed from $\y$ and $\x$ are known. We determine a model of $\f$ so that an associated error is minimized. To this end, the model $\ttt_p$ is constructed as a sum of $p+1$ particular parts, in the form $\ttt_p (\y) = \sum_{j=0}^{p}G_j H_j Q_j(\vv_j)$ where $G_j$ and $ H_j$, for $j=0,\ldots, p$, are matrices to be determined, and $\vv_j$, for $j=1,\ldots,p$, is a special random vector called the injection. We denote $\vv_0=\y$. Injections $\vv_1,\ldots, \vv_p$ are aimed to diminish the error associated with the proposed model $\ttt_p$. Further, $Q_j$ is a special transform aimed to facilitate the numerical realization of model $\ttt_p$. It is determined in the way allowing us to optimally determine $G_j$ and $ H_j$ as a solution of $p+1$ separate error minimization problems which are simpler than the original minimization problem. The empirical determination of injections $\vv_1,\ldots, \vv_p$ is considered. The proposed method has several degrees of freedom to diminish the associated error. They are `degree' $p$ of $\ttt_p$, choice of matrices $G_0, H_0, \ldots,$ $G_p, H_p$, dimensions of matrices $G_0, H_0, \ldots,$ $G_p, H_p$ and injections $\vv_1,\ldots, \vv_p$, respectively. Four numerical examples are provided. At the end, the open problem is formulated.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 JUAN PABLO SOTO QUIROS, Anatoli Torokhtihttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5805Monophonic-triangular Distance in Graphs2022-12-23T12:12:43+00:00P.Ttius P.titusvino@yahoo.com<p>A path u1, u2, ..., un in a connected graph G such that for i, j with j ≥ i + 3, there does not exist an edge uiuj , is called a monophonic-triangular path or mt-path. The monophonic-triangular distance or mt-distance dmt(u, v) from u to v is defined as the length of a longest u−v mt-path in G. The mt-eccentricity emt(v) of a vertex v in G is defined as the maximum mt-distance between v and other vertices in G. The mt-radius radmt(G) is defined as the minimum mt-eccentricity among the vertices of G and the mt-diameter diammt(G) is defined as the maximum mt-eccentricity among the vertices of G. It is shown that radmt(G) ≤ diammt(G) for every connected graph G. Some realization and characterization results are given based on mt-radius, mt-diameter, mt-center and mt-periphery of a connected graph.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 P.Ttius P.https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5594Investigating Banhatti indices on the molecular graph and the line graph of Glass with M- polynomial approach 2023-06-21T15:16:07+00:00masoud ghodsmghods@semnan.ac.irJaber Ramezani Tousijaber.ramezani@semnan.ac.ir<p>Topological indices are numerical values related to a chemical structure that describes the correlation of chemical structure with different physical properties and chemical reactions. Glass has wide applications in architecture, tableware, optics, and optoelectronics.</p> <p>In this article, first, the mathematical relationship between M-polynomial and Banhatti indices such as K-Banhatti, d-Banhatti, and hyper d-Banhatti indices are obtained. Then using M-polynomial, Banhatti indices are calculated.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 masoud ghods; Jaber Ramezani Tousihttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/6175Further results on edge irregularity strength of some graphs2023-10-16T14:39:32+00:00Muhammad Imranimranbepakistani@gmail.comMurat Cancanmcancan@yyu.edu.trMuhammad Faisal Nadeemmfaisalnadeem@ymail.comYasir Aliyasirbepakistani@gmail.com<p>The focal point of this paper is to precisely ascertain the edge irregularity strength of various finite, simple, and undirected captivating graphs, including splitting graph, shadow graph, jewel graph, jellyfish graph, and $m$ copies of 4-pan graph.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 Muhammad Imran, Murat Cancan, Muhammad Faisal Nadeem, Yasir Alihttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5541Nourishing Number of Some Associated Graphs2022-10-26T15:31:23+00:00Udayan Prajapatiudayan64@yahoo.comKishan Vyaskishanvyas440@gmail.com<p>Let N<sub>0</sub> = N∪{0} and P(N<sub>0</sub>) be the power set. An injection f : V (G) → P(N<sub>0</sub>) is an integer additive set-indexer (IASI) of a graph G if the induced map f<sup>+</sup> : E(G) → P(N<sub>0</sub>) given by f<sup>+</sup>(uv) = f(u) + f(v) is also an injection, where f(u) + f(v) is the sumset of f(u) and f(v). Moreover, if |f<sup>+</sup>(uv)| = |f(u)| |f(v)|, for all uv in E(G), then f is a strong IASI of G. The nourishing number of a graph G is the minimum order of the maximal complete subgraph of G such that G admits a strong IASI. In this paper we investigate the admissibility of strong IASI for some associated graphs<br>and calculate their nourishing number. In addition, we obtain the nourishing number of powers of the associated graphs.</p>2024-03-07T00:00:00+00:00Copyright (c) 2024 Udayan Prajapati, Mr. Kishan Vyashttps://www.revistaproyecciones.cl/index.php/proyecciones/article/view/6047Projective non-commuting graph of a group2023-06-08T07:06:59+00:00Julio Cesar Moraes Pezzottjuliopezzott@gmail.com<p>Let $G$ be a finite non-abelian group and let $T$ be a transversal of the center of $G$ in $G$. <br>The non-commuting graph of $G$ on a transversal of the center is the graph whose vertices are the non-central elements of $T$ and two vertices $x$ and $y$ are joined by an edge whenever $xy \neq yx$. In this paper, we classify the groups whose non-commuting graph on a transversal of the center is projective.</p>2024-03-11T00:00:00+00:00Copyright (c) 2024 Julio Cesar Moraes Pezzott