Proyecciones (Antofagasta, On line) https://www.revistaproyecciones.cl/index.php/proyecciones <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 <div id="deed-conditions" class="row"> <ul class="license-properties col-md-offset-2 col-md-8" dir="ltr"> <li class="license by"> <p><strong>Attribution</strong> — You must give <a id="appropriate_credit_popup" class="helpLink" tabindex="0" title="" href="https://creativecommons.org/licenses/by/4.0/deed.en" data-original-title="">appropriate credit</a>, provide a link to the license, and <a id="indicate_changes_popup" class="helpLink" tabindex="0" title="" href="https://creativecommons.org/licenses/by/4.0/deed.en" data-original-title="">indicate if changes were made</a>. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.<span id="by-more-container"></span></p> </li> </ul> </div> <div class="row"> <ul id="deed-conditions-no-icons" class="col-md-offset-2 col-md-8"> <li class="license"><strong>No additional restrictions</strong> — You may not apply legal terms or <a id="technological_measures_popup" class="helpLink" tabindex="0" title="" href="https://creativecommons.org/licenses/by/4.0/deed.en" data-original-title="">technological measures</a> that legally restrict others from doing anything the license permits.</li> </ul> </div> Commuting graph of CA−groups https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4488 <p>A group G is called a CA−group, if all the element centralizers of G are abelian and the commuting graph of G with respect to a subset A of G, denoted by Γ(G, A), is a simple undirected graph with vertex set A and two distinct vertices a and b are adjacent if and only if ab = ba. The aim of this paper is to generalize results of a recently published paper of F. Ali, M. Salman and S. Huang [On the commuting graph of dihedral group, Comm. Algebra 44 (6) (2016) 2389—2401] to the case that G is an CA−group.</p> Mehdi Torktaz Ali Reza Ashrafi Copyright (c) 2023 Mehdi Torktaz, Ali Reza Ashrafi https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 1 17 10.22199/issn.0717-6279-4488 Mappings preserving sum of products a◊b + b*a (resp., a*◊b + ab*) on ∗-algebras https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5139 <p>Let A and B be two prime complex ∗-algebras. We proved that every bijective mapping Φ : A → B satisfying Φ(a ◊+ b<sup>∗ </sup>a) = Φ(a)◊Φ(b) + Φ(b)<sup>∗</sup>Φ(a) (resp., Φ(a<sup>∗ </sup>◊b + ab<sup>∗</sup>) = Φ(a)<sup>∗</sup> ◊Φ(b) + Φ(a)Φ(b)<sup>∗</sup>), where a ◊b = ab + ba<sup>∗</sup>, for all elements a, b ∈ A, is a ∗-ring isomorphism.</p> Ali Taghavi João Carlos Ferreira Maria das Graças Marietto Copyright (c) 2023 Ali Taghavi, João Carlos Ferreira, Maria das Graças Marietto https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 18 31 10.22199/issn.0717-6279-5139 Open global shadow graph and it’s zero forcing number https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5305 <p>Zero forcing number of a graph is the minimum cardinality of the zero forcing set. A zero forcing set is a set of black vertices of minimum cardinality that can colour the entire graph black using the color change rule: each vertex of G is coloured either white or black, and vertex v is a black vertex and can force a white neighbour only if it has one white neighbour. In this paper we identify a class of graph where the zero forcing number is equal to the minimum rank of the graph and call it as a new class of graph that is open global shadow graph”. Some of the basic properties of open global shadow graph are studied. The zero forcing number of open global shadow graph of a graph with upper and lower bound is obtained. Hence giving the upper and lower bound for the minimum rank of the graph.</p> Raksha Malur Charles Dominic Copyright (c) 2023 Raksha Malur, Charles Dominic https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 33 52 10.22199/issn.0717-6279-5305 Jordan product and fixed points preservers https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5375 <p>Let B(X) be the space of all bounded linear operators on complex Banach space X. For A ∈ B(X), we denote by F(A) the subspace of all fixed points of A. In this paper, we study and characterize all surjective maps φ on B(X) satisfying</p> <p>F(φ(T)φ(A) + φ(A)φ(T)) = F(T A + AT)</p> <p>for all A, T ∈ B(X).</p> <pre> </pre> Mhamed Elhodaibi Soufiane Elouazzani Copyright (c) 2023 Mhamed Elhodaibi, Soufiane Elouazzani https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 53 63 10.22199/issn.0717-6279-5375 A note on m-Zumkeller cordial labeling of graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5190 <p>Let G(V,E) be a graph. An m-Zumkeller cordial labeling of the graph G is defined by an injective function f:V -&gt; N such that there exists an induced function f*:E --&gt;{0,1} defined by f* (uv)=f(u).f(v) that satisfies the following conditions:<br /><br />i) For every uv in E,</p> <p>f*(uv)= <img src="https://www.revistaproyecciones.cl/public/site/images/rvidal/imagen1-5.png" alt="" width="390" height="38" /> <br /><br />ii) |e<sub>f*(0)</sub>-e<sub>f*(1)</sub>|&lt;=1<br /><br />where e<sub>f*(0)</sub> and e<sub>f*(1)</sub> denote the number of edges of the graph G having label 0 and 1 respectively under f*.<br /><br />In this paper we prove that there exist an m -Zumkeller cordial labeling of graphs viz., (i) paths (ii) cycles (iii) comb graphs (iv) ladder graphs (v) twig graphs (vi) helm graphs (vii) wheel graphs (viii) crown graphs (ix) star graphs.</p> Harish Patodia Helen K. Saikia Copyright (c) 2023 Harish Patodia, Helen K. Saikia https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 65 84 10.22199/issn.0717-6279-5190 Properties of nearly S-paracompact spaces https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5482 <p>We study some basic properties of a nearly S-paracompact space and its characterizations under certain hypotheses about space. We establish relationships between this class of spaces and other well-known spaces. Also, we analyze the invariance of nearly S-paracompactness under direct and inverse images of some types of functions.</p> José Eduardo Sanabria Ennis Rosas Clara Blanco Copyright (c) 2023 José Eduardo Sanabria, Ennis Rosas, Clara Blanco https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 91 103 10.22199/issn.0717-6279-5482 Spectra of (M, ℳ)-corona-join of graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5454 <p>In this paper, we introduce the (M, ℳ)-corona-join of G and ℋ<sub>k</sub> constrained by vertex subsets 𝒯, which is the union of two graphs: one is the M-generalized corona of a graph G and a family of graphs ℋ<sub>k</sub> constrained by vertex subset 𝒯 of the graphs in ℋ<sub>k</sub>, where M is a suitable matrix; and the other one is the ℳ -join of ℋ<sub>k</sub>, where ℳ is a collection of matrices. We determine the spectra of the adjacency, the Laplacian, the signless Laplacian and the normalized Laplacian matrices of some special cases of the (M, ℳ)-corona-join of G and ℋ<sub>k</sub> constrained by vertex subsets 𝒯. These results enable us to deduce the spectra of all the existing variants of extended corona of graphs. Further, by using this graph operation, we construct infinitely many graphs which are simultaneously cospectral with respect to the above mentioned four type of matrices.</p> Gayathri Murugesan Rajkumar Rajendran Copyright (c) 2023 M. Gayathri, R. Rajkumar https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 105 124 10.22199/issn.0717-6279-5454 On extended biharmonic hypersurfaces with three curvatures https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5443 <p>The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics and differential geometry, arises from the variation problems of ordinary mean curvature vector field. Generally, harmonic submanifolds are biharmonic, but not vice versa. Of course, many examples of biharmonic hypersurfaces are harmonic. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every biharmonic submanifold is harmonic. Although the conjecture has not been proven (in general case), it has been affirmed in many cases, and this has led to its spread to various types of submanifolds. Inspired by the conjecture, we study the Lorentz submanifolds of the Lorentz-Minkowski spaces. We consider an advanced versión of the conjecture (namely, L<sub>1</sub>-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean 5-space <strong>L<sup>5</sup></strong> := <em>E</em><sub>1</sub><sup>5</sup> (i.e. the Minkowski 5-space). We confirm the extended conjecture on Lorentz hypersurfaces with three principal curvatures.</p> F. Pashaie N. Tanoomand-Khooshmehr A. Rahimi L. Shahbaz Copyright (c) 2023 F. Pashaie, N. Tanoomand-Khooshmehr, A. Rahimi, L. Shahbaz https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 125 144 10.22199/issn.0717-6279-5443 Resistance distance of generalized wheel and dumbbell graph using symmetric {1}-inverse of Laplacian matrix https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5510 <p>A new class of graphs called dumbbell graphs, denoted by <strong>DB</strong>(Wm,n) is the graph obtained from two copies of generalized wheel graph W<sub>m,n, </sub>m ≥ 2, n ≥ 3. It is a graph on 2 (m + n) vertices obtained by connecting m-vertices in one copy with the corresponding vertices in the other copy. The resistance distance between two vertices v<sub>i</sub> and v<sub>j</sub>, denoted by r<sub>ij</sub> , is defined as the effective electrical resistance between them if each edge of G is replaced by 1 ohm resistor. The Kirchhoff index is the sum of the resistance distances between all pairs of vertices in the graph. In this paper, we formulate the resistance distance of W<sub>m,n </sub>and <strong>DB</strong>(W<sub>m,n</sub>) using Symmetric {1}-inverse of Laplacian matrices. We provide examples to illustrate the proposed method and also obtain the Kirchhoff indices for these examples.</p> Sakthidevi Kaliyaperumal Kalyani Desikan Copyright (c) 2023 Sakthidevi Kaliyaperumal, Kalyani Desikan https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 145 166 10.22199/issn.0717-6279-5510 A note on fold thickness of graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5655 <p>A 1-fold of G is the graph G<sup>0</sup> obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor and reducing the resulting multiple edges to simple edges. A uniform k-folding of a graph G is a sequence of graphs</p> <p>G = G<sub>0</sub>, G<sub>1</sub>, G<sub>2</sub>,...,G<sub>k</sub>, where G<sub>i+1</sub> is a 1-fold of G<sub>i</sub> for</p> <p>i = 0, 1, 2,...,k − 1 such that all graphs in the sequence are singular or all of them are nonsingular. The largest k for which there exists a uniform k- folding of G is called fold thickness of G and this concept was first introduced in [1]. In this paper, we determine fold thickness of corona product graph G ʘ K<sub>m </sub>, G ʘ <sub>S</sub> <sub> </sub>, K<sub>m</sub>and graph join G + K<sub>m</sub> <sub> </sub>.</p> T. Reji S. Vaishnavi Francis Joseph H. Campeña Copyright (c) 2023 T. Reji, S. Vaishnavi, Francis Joseph H. Campeña https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 167 174 10.22199/issn.0717-6279-5655 Chromatic coloring of distance graphs, III https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5689 <p>A graph G(Z, D) with vertex set Z is called an integer distance graph if its edge set is obtained by joining two elements of Z by an edge whenever their absolute difference is a member of D. When D = P or D ⊆ P where P is the set of all prime numbers then we call it a prime distance graph. After establishing the chromatic number of G(Z, P ) as four, Eggleton has classified the collection of graphs as belonging to class i if the chromatic number of G(Z, D) is i. The problem of characterizing the family of graphs belonging to class i when D is of any given size is open for the past few decades. As coloring a prime distance graph is equivalent to producing a prime distance labeling for vertices of G, we have succeeded in giving a prime distance labeling for certain class of all graphs considered here. We have proved that if D = {2, 3, 5, 7, 7<sup>th</sup> prime, 10<sup>th</sup> prime, 13<sup>th</sup> prime, 16<sup>th</sup> prime, (7 +<img src="https://www.revistaproyecciones.cl/public/site/images/rvidal/imagen1.png" alt="" width="70" height="24" /> <sup>j</sup>)<sup>th</sup> prime, ..., (4 + <img src="https://www.revistaproyecciones.cl/public/site/images/rvidal/imagen2.png" alt="" width="69" height="24" /><sup>j</sup>)<sup>th</sup> prime for any s ∈ N}, then there exists a prime distance graph with distance set D in class 4 and if D = {2, 3, 5, 4<sup>th</sup> prime, 6<sup>th</sup> prime, 8<sup>th</sup> prime, (4 + <img src="https://www.revistaproyecciones.cl/public/site/images/rvidal/imagen3.png" alt="" width="69" height="24" /> <sup>j</sup>)<sup>th</sup> prime, ..., (2 + <img src="https://www.revistaproyecciones.cl/public/site/images/rvidal/imagen4.png" alt="" width="69" height="24" /><sup>j</sup>)<sup>th</sup> prime for any s ∈ N} then there exists a prime distance graphs with distance set D in class 3. Further, we have also obtained some more interesting results that are either general or existential such as a) If D is a specific sequence of integers in arithmetic progression then there exist a prime distance graph with distance set D, b) If G is any prime distance graph in class i for 1 ≤ i ≤ 4 then G × K<sub>2</sub> is also a prime distance graph in the respective class i, c) A countable union of disjoint copies of prime distance graph is again a prime distance graph, d) The Middle/Total graph of a path on n vertices is a prime distance graph. In addition we also provide a new different proof for establishing a fact that all cycles are prime distance graph.</p> George Barnabas Venkataraman Yegnanarayanan Copyright (c) 2023 George Barnabas, Venkataraman Yegnanarayanan https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 175 204 10.22199/issn.0717-6279-5689 On Randić energy of graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4330 <p>Let d<sub>i</sub> be the degree of vertex vi of G then Randić matrix R(G) = [r<sub>ij </sub>] is defined as r<sub>ij</sub> = 1/ √d<sub>i</sub>d<sub>j</sub>, if the vertices vi and vj are adjacent in G or r<sub>ij</sub> = 0, otherwise. The Randić energy is the sum of absolute values of the eigenvalues of R(G). In this paper we have investigated Randić energy of m-Splitting and m-Shadow graphs. We also have constructed a sequence of graphs having same Randić energy.</p> Samir K. Vaidya G. K. Rathod Copyright (c) 2023 Samir K. Vaidya, G. K. Rathod https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 205 218 10.22199/issn.0717-6279-4330 Study of multiplicative derivation and its additivity https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5578 <p>In this paper, we modify the result of M. N. Daif [1] on multiplicative derivations in rings. He showed that the multiplicative derivation is additive by imposing certain conditions on the ring ℜ. Here, we have proved the above result with lesser conditions than M. N. Daif for getting multiplicative derivation to be additive.</p> Wasim Ahmed Muzibur Rahman Mozumder Arshad Madni Copyright (c) 2023 Wasim Ahmed, Dr. Muzibur Rahman Mozumder, Arshad Madni https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 219 232 10.22199/issn.0717-6279-5578 A graph product and its applications in generating non-cospectral equienergetic graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5489 <p>A new graph product is defined in this paper and several applications of this product are described. The adjacency matrix of the product graph is given and its complete spectrum in terms of the spectrum of constituent graphs are determined. Sequences of cospectral graphs can be generated from the known cospectral graphs using the new product. Several sequences of non-cospectral equienergtic graphs can also be generated as an application of the graph product defined.</p> Subin P. Joseph Copyright (c) 2023 Subin P. Joseph https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 233 244 10.22199/issn.0717-6279-5489 A correction on “Some remarks on fuzzy infi-topological spaces” https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5430 <p>In [3], the authors present a weak fuzzy topological structure called “fuzzy infi topological space” and studied some of its consequences. In this paper, we show that some of the results in [3] are incorrect and provide the correct versions of them.</p> Zanyar Anwer Ameen Copyright (c) 2023 Zanyar Anwer Ameen https://creativecommons.org/licenses/by/4.0 2023-01-26 2023-01-26 42 1 85 90 10.22199/issn.0717-6279-5430