Serdica Mathematical Journal

Volume 46, Number 1, 2020

C O N T E N T S

Survey

· Valov, V. Homogeneous metric ANR-compacta (pp. 1−18)

Articles

· Zagorodnyuk, S. M. On extensions of commuting tuples of symmetric and isometric operators (pp. 19−34)
· Naveenkumar, S. H. Result on uniqueness and weighted sharing of entire function of differential polynomial with their derivatives (pp. 35−42)
· Barman, S. C., M. Pal, S. Mondal. An optimal algorithm for computing minimum k-hop dominating set of permutation graphs (pp. 43−60)
· Reddy, B. S., R. S. Jain, N. Laxmikanth. Characteristic polynomial and Wiener index of the compressed zero divisor graph Γ_E[Z_pⁿ] (pp. 61−72)
· Ebrahimi Atani, S., M. Chenari. Supplemented property in the lattices (pp. 73−88)
· Mittal, G. Non-trivial idempotents of the matrix rings over the polynomial ring Zpqr[x] (pp. 89−100)

A B S T R A C T S

HOMOGENEOUS METRIC ANR-COMPACTA
V. Valov veskov@nipissingu.ca

2020 Mathematics Subject Classification: Primary 54C55, Secondary 55M15.
Key words: absolute neighborhood retracts, cohomological dimension, cohomology and homology groups, homogeneous compacta.

This is a survey of most important results and unsolved problems about homogeneous finite-dimensional metric ANR-compacta. We also discuss some partial results and possible ways of solutions.

ON EXTENSIONS OF COMMUTING TUPLES OF SYMMETRIC AND ISOMETRIC OPERATORS
Sergey M. Zagorodnyuk Sergey.M.Zagorodnyuk@gmail.com, Sergey.M.Zagorodnyuk@univer.kharkov.ua

2020 Mathematics Subject Classification: 47A20.
Key words: extensions of operators, symmetric operators, isometric operators, moment problems.

In this paper we study extensions of commuting tuples of symmetric and isometric operators to commuting tuples of self-adjoint and unitary operators. Some conditions which ensure the existence of such extensions are presented. A multidimensional analog of the Godič−Lucenko Theorem is proved. An application to a multidimensional power-trigonometric moment problem is given.

2020 Mathematics Subject Classification: Primary 30D35.
Key words: entire function, uniqueness, small function, weighted sharing, differential polynomials.

In this article, we worked on the uniqueness of more generalised form of a function namely fⁿP(f) and [fⁿP(f)]^(k) sharing a small function. The Theorem proved in this paper improves the results of Kit-wing [9] (On entire and meromorphic functions that share small functions with their derivatives. JIPAM. J. Inequal. Pure Appl. Math. 4, No 1 (2003), Article 21, 7 pp).

AN OPTIMAL ALGORITHM FOR COMPUTING MINIMUM k-HOP DOMINATING SET OF PERMUTATION GRAPHS
Sambhu Charan Barman barman.sambhu@gmail.com,
Madhumangal Pal mmpalvu@gmail.com,
Sukumar Mondal sm5971@rediffmail.com

2020 Mathematics Subject Classification: 05C30, 05C12, 68R10, 68Q25.
Key words: Design & analysis of algorithms, k-hop domination, permutation graphs.

A k-hop dominating set (k-HDS) D of a graph G = (V,E) is a subset of V such that every vertex x ∈ V is within k-steps from at least one vertex y ∈ D, i.e., d(x,y)≤ k, where k is a fixed positive integer. A k-hop dominating set $D$ is said to be minimal if there does not exist any H ⊂ D such that H is a k-HDS of G. If a dominating set D is minimal as well as it is k-HDS then it is said to be minimum k-hop dominating set. In this paper, we present an optimal algorithm to compute a minimum k-HDS of permutation graphs with n vertices which runs in O(n) time.

CHARACTERISTIC POLYNOMIAL AND WIENER INDEX OF THE COMPRESSED ZERO DIVISOR GRAPH Γ_E[Z_pⁿ]
B. Surendranath Reddy surendra.phd@gmail.com,
Rupali S. Jain rupalisjain@gmail.com
N. Laxmikanth laxmikanth.nandala@gmail.com

2020 Mathematics Subject Classification: Primary: 13A70; Secondary: 05C12, 05C25, 13A15.
Key words: compressed zero divisor graph, characteristic polynomial, binomial coefficients, Wiener index.

The zero divisor graph of a commutative ring R, denoted by Γ[R], is a graph whose vertices are non-zero zero divisors of R and two vertices are adjacent if their product is zero. The relation on R given by r ∽ s if and only if ann_R(r) = ann_R(s) is an equivalence relation. The compressed zero divisor graph Γ_E(R) is the (undirected) graph whose vertices are the equivalence classes induced by ∽ other than [0] and [1], such that distinct vertices [r] and [s] are adjacent in Γ_E(R) if and only if rs = 0. We show that the coefficients of the compressed zero divisor graph Γ_E[Z_pⁿ] form a Pascal like triangle and thus derive the characteristic polynomial in terms of binomial coefficients. We also calculate the Wiener index of Γ_E[Z_pⁿ].

SUPPLEMENTED PROPERTY IN THE LATTICES
Shahabaddin Ebrahimi Atani ebrahimiatani@gmail.com,
Maryam Chenari chenari.maryam13@gmail.com

2020 Mathematics Subject Classification: 06C05, 06C15.
Key words: lattices, supplemented filters, amply supplemented filters, hollow filter.

Let L be a lattice with the greatest element 1. In a manner analogous to a module over a ring, we introduce (amply) supplemented filters of L. The basic properties and possible structures of such filters are investigated.

NON-TRIVIAL IDEMPOTENTS OF THE MATRIX RINGS OVER THE POLYNOMIAL RING Zpqr[x]
Gaurav Mittal gmittal@ma.iitr.ac.in

2020 Mathematics Subject Classification: 16S50, 13F20.
Key words: idempotent, polynomial ring, matrix ring.

In this paper, we study the non-trivial idempotents of the 2 × 2 matrix ring over the polynomial ring Z_pqr[x] for distinct primes p, q and r greater than 3. We have classified all the idempotents of this matrix ring into several classes such that any idempotent must belong to one of these classes. This work is extension of the work done in [1] (J. M. P. Balmaceda, J. P. P. Datu. Idempotents in certain matrix rings over polynomial rings. Int. Electron. J. Algebra 27 (2020), 1−12).