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

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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]
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Database: Engineering Source
Description
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]
ISSN:00243795
DOI:10.1016/j.laa.2025.11.009