View in EDS

The P-vertex problem for symmetric matrices whose associated graphs admit perfect matchings.

Saved in:
Bibliographic Details
Title: The P-vertex problem for symmetric matrices whose associated graphs admit perfect matchings.
Authors: Sharma, K.1 (AUTHOR) kshitijsharma.pi@kgpian.iitkgp.ac.in, Panda, S.K.1 (AUTHOR)
Source: Linear Algebra & its Applications. Feb2026, Vol. 731, p109-138. 30p.
Subjects: Symmetric matrices, Bipartite graphs, Matching theory, Graph theory
Abstract: Let G be the underlying graph of a real symmetric matrix A. Denote by A (j) the principal submatrix of A obtained by deleting the j th row and column, and let m A (λ i) denote the algebraic multiplicity of the eigenvalue λ i of A. An index j is called a P-vertex of A if m A (j) (0) − m A (0) = 1. A graph G on n vertices is said to have property (P) if there exists a nonsingular symmetric matrix A whose underlying graph is G such that every vertex of A is a P-vertex. This work develops a graph-theoretic framework for studying property (P), with particular emphasis on graphs that admit a perfect matching. We analyze bipartite graphs that satisfy property (P) and show that the existence of a perfect matching plays a decisive role in their characterization. In particular, we prove that a tree possesses property (P) if and only if it admits a unique perfect matching, and we present an alternative characterization of unicyclic graphs satisfying property (P). The analysis is then extended to non-bipartite graphs with a unique perfect matching, where we highlight structural features that influence property (P). Furthermore, we construct a family of graphs on n vertices that do not satisfy property (P), but have a nonsingular matrix for which the number of P-vertices is n − 1. We also investigate the behavior of property (P) under certain graph operations, such as the edge-joining of graphs, and show that this operation preserves property (P) under specific conditions. In particular, we establish that if two graphs G and H each satisfy property (P), then the graph obtained by joining them with a single edge also satisfies property (P), and we examine the converse of this result. [ABSTRACT FROM AUTHOR]
Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
FullText Text:
  Availability: 0
Header DbId: egs
DbLabel: Engineering Source
An: 189790684
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: The P-vertex problem for symmetric matrices whose associated graphs admit perfect matchings.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Sharma%2C+K%2E%22">Sharma, K.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> kshitijsharma.pi@kgpian.iitkgp.ac.in</i><br /><searchLink fieldCode="AR" term="%22Panda%2C+S%2EK%2E%22">Panda, S.K.</searchLink><relatesTo>1</relatesTo> (AUTHOR)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22Linear+Algebra+%26+its+Applications%22">Linear Algebra & its Applications</searchLink>. Feb2026, Vol. 731, p109-138. 30p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Symmetric+matrices%22">Symmetric matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Bipartite+graphs%22">Bipartite graphs</searchLink><br /><searchLink fieldCode="DE" term="%22Matching+theory%22">Matching theory</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Let G be the underlying graph of a real symmetric matrix A. Denote by A (j) the principal submatrix of A obtained by deleting the j th row and column, and let m A (λ i) denote the algebraic multiplicity of the eigenvalue λ i of A. An index j is called a P-vertex of A if m A (j) (0) − m A (0) = 1. A graph G on n vertices is said to have property (P) if there exists a nonsingular symmetric matrix A whose underlying graph is G such that every vertex of A is a P-vertex. This work develops a graph-theoretic framework for studying property (P), with particular emphasis on graphs that admit a perfect matching. We analyze bipartite graphs that satisfy property (P) and show that the existence of a perfect matching plays a decisive role in their characterization. In particular, we prove that a tree possesses property (P) if and only if it admits a unique perfect matching, and we present an alternative characterization of unicyclic graphs satisfying property (P). The analysis is then extended to non-bipartite graphs with a unique perfect matching, where we highlight structural features that influence property (P). Furthermore, we construct a family of graphs on n vertices that do not satisfy property (P), but have a nonsingular matrix for which the number of P-vertices is n − 1. We also investigate the behavior of property (P) under certain graph operations, such as the edge-joining of graphs, and show that this operation preserves property (P) under specific conditions. In particular, we establish that if two graphs G and H each satisfy property (P), then the graph obtained by joining them with a single edge also satisfies property (P), and we examine the converse of this result. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=189790684
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1016/j.laa.2025.11.009
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 30
        StartPage: 109
    Subjects:
      – SubjectFull: Symmetric matrices
        Type: general
      – SubjectFull: Bipartite graphs
        Type: general
      – SubjectFull: Matching theory
        Type: general
      – SubjectFull: Graph theory
        Type: general
    Titles:
      – TitleFull: The P-vertex problem for symmetric matrices whose associated graphs admit perfect matchings.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Sharma, K.
      – PersonEntity:
          Name:
            NameFull: Panda, S.K.
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 15
              M: 02
              Text: Feb2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 00243795
          Numbering:
            – Type: volume
              Value: 731
          Titles:
            – TitleFull: Linear Algebra & its Applications
              Type: main
ResultId 1