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Complement of the reduced non-zero component graph of free semimodules

Complement of the reduced non-zero component graph of free semimodules Let M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{M}$$\end{document} be a finitely generated free semimodule over a semiring S\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{S}$$\end{document} with identity having invariant basis number property with a basis α = {α1,…, αk}. The complement Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document} of the reduced non-zero component graph Γ∗(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\Gamma ^ * }\left(\mathbb{M}\right)$$\end{document} of M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{M}$$\end{document}, is the simple undirected graph with V=M∗\{∑i=1kciαi:ci≠0∀i}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V = {\mathbb{M}^ * }\backslash \left\{ {\sum\limits_{i = 1}^k {{c_i}} {\alpha _i}:{c_i} \ne 0\,\,\forall \,\,i} \right\}$$\end{document} as the vertex set and such that there is an edge between two distinct vertices a=∑i=1kaiαi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a = \sum\limits_{i = 1}^k {{a_i}{\alpha _i}} $$\end{document} and b=∑i=1kbiαi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b = \sum\limits_{i = 1}^k {{b_i}{\alpha _i}} $$\end{document} if and only if there exists no i such that both ai, bi are non-zero. In this paper, we show that the graph Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document} is connected and find its domination number, clique number and chromatic number. In the case of finite semirings, we determine the degree of each vertex, order, size, vertex connectivity and girth of Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document}. Also, we give a necessary and sufficient condition for Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document} to be Eulerian or Hamiltonian or planar. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics-A Journal of Chinese Universities Springer Journals

Complement of the reduced non-zero component graph of free semimodules

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Publisher
Springer Journals
Copyright
Copyright © Editorial Committee of Applied Mathematics 2023
ISSN
1005-1031
eISSN
1993-0445
DOI
10.1007/s11766-023-3737-5
Publisher site
See Article on Publisher Site

Abstract

Let M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{M}$$\end{document} be a finitely generated free semimodule over a semiring S\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{S}$$\end{document} with identity having invariant basis number property with a basis α = {α1,…, αk}. The complement Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document} of the reduced non-zero component graph Γ∗(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\Gamma ^ * }\left(\mathbb{M}\right)$$\end{document} of M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{M}$$\end{document}, is the simple undirected graph with V=M∗\{∑i=1kciαi:ci≠0∀i}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V = {\mathbb{M}^ * }\backslash \left\{ {\sum\limits_{i = 1}^k {{c_i}} {\alpha _i}:{c_i} \ne 0\,\,\forall \,\,i} \right\}$$\end{document} as the vertex set and such that there is an edge between two distinct vertices a=∑i=1kaiαi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a = \sum\limits_{i = 1}^k {{a_i}{\alpha _i}} $$\end{document} and b=∑i=1kbiαi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b = \sum\limits_{i = 1}^k {{b_i}{\alpha _i}} $$\end{document} if and only if there exists no i such that both ai, bi are non-zero. In this paper, we show that the graph Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document} is connected and find its domination number, clique number and chromatic number. In the case of finite semirings, we determine the degree of each vertex, order, size, vertex connectivity and girth of Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document}. Also, we give a necessary and sufficient condition for Γ∗¯(M)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline {{\Gamma ^ * }} \left(\mathbb{M}\right)$$\end{document} to be Eulerian or Hamiltonian or planar.

Journal

Applied Mathematics-A Journal of Chinese Universities – Springer Journals

Published: Mar 1, 2023

Keywords: semiring; modules; planar; 16Y60; 06F25; 05C10

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