arXiv subjects

Puja Samanta

Publications and source records attributed to Puja Samanta.

3 recordsLinked to original sources

arXiv2026

The Cycle Rank Threshold: Perfect Matchings and Bipartite Parter Graphs

A graph on $n$ vertices is called a Parter graph if there exists a nonsingular symmetric matrix, whose nonzero off-diagonal entries correspond exactly to the edges of the graph, such that all of its principal submatrices of order $n-1$ are singular. Previously, a graph satisfying this condition was said to have property~(P). It was proved that, for bipartite graphs of cycle rank at most $3$, being a Parter graph is equivalent to the existence of a perfect matching. We extend this result to cycle rank $4$, proving that every bipartite graph of cycle rank at most $4$ is a Parter graph if and only if it has a perfect matching. Furthermore, we show that this bound is sharp by constructing, for every integer $r\ge5$, a connected balanced bipartite Parter graph of cycle rank $r$ that has no perfect matching.

math.CO↗

arXiv2026

The Full P-vertex Problem and Perfect Matchings for Bipartite Graphs

In a recent work, Sharma and Panda~\cite{sharma} showed that every bipartite graph with a perfect matching has property (P), and proved the converse for trees and unicyclic bipartite graphs (i.e., bipartite graphs with cycle rank $m(G) \le 1$). In this paper, we extend this result to broader classes of bipartite graphs. We first show that every bipartite graph with property (P) is balanced. We prove that the converse holds for all bipartite graphs with cycle rank $m(G)$ at most three, and further establish it for several additional families of bipartite graphs. Finally, we derive algebraic constraints for balanced bipartite graphs without perfect matchings and use them to identify a family of bipartite graphs that does not have property (P).

math.CO↗

arXiv2026

Zero-sum Inverse Realization and Property~(P) under Join Operations

We introduce zero-sum inverse realization of property (P) of a graph $G$, obtained by imposing an additional condition $ \mathbf{1}^{\top}A^{-1}\mathbf{1}=0, $ on a matrix $A\in S(G)$ realizing property (P), where $\mathbf{1}$ is the all-ones vector. We prove that every graph of order at least three having property (P) admits such a zero-sum inverse realization. As applications, we prove that property~(P) is preserved under the join of two graphs of order at least $3$ and, more generally, under the $H$-join of a family of graphs of order at least $3$, where $H$ is arbitrary. Consequently, we obtain sufficient conditions for cographs and lexicographic products of graphs to possess property~(P). Throughout the paper, many examples are given.

math.CO↗