arXiv subjects

Priyanshu Pant

Publications and source records attributed to Priyanshu Pant.

5 recordsLinked to original sources

arXiv2026

Exponential Lower Bounds for the Pfaffian Number of Graphs

The Fisher--Kasteleyn--Temperley (FKT) algorithm counts perfect matchings in planar graphs in polynomial time using a single Pfaffian computation. Galluccio--Loebl and Tesler extended this Pfaffian method to graphs embedded in an orientable surface of genus $g$, showing that the perfect-matching polynomial can be written as a linear combination of at most $4^g$ Pfaffians. We prove that this exponential dependence on $g$ is unavoidable in general. More precisely, for every $g\ge1$, there exists a graph of orientable genus at most $g$ whose perfect-matching polynomial requires at least $(8/3)^g$ Pfaffians in any such linear representation. In particular, for every even integer $n\ge 6$, there is a graph on $n$ vertices with Pfaffian number at least $(8/3)^{\lfloor n/6\rfloor}$. Moreover, the lower bound is witnessed even by connected cubic bipartite matching-covered graphs of orientable genus exactly $g$. We also prove exponential lower bounds for complete bipartite graphs, and hence for even complete graphs, improving asymptotically on a recent linear lower bound of Junchaya, Miranda, and Lucchesi.

math.CO↗

arXiv2026

Counterexamples to a Conjecture on Laplacian Ratios of Trees

For a graph $G$ with no isolated vertices, its Laplacian ratio is defined as \[ π(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where $L(G)$ is the Laplacian matrix of $G$, $d(v)$ is the degree of $v$, and $\operatorname{per}$ denotes the permanent. Brualdi and Goldwasser asked for the maximum value of $π(T)$ among trees $T$ with a fixed number of vertices. Wu, Dong and Lai recently proposed a conjectural answer to this problem. We give infinite families of counterexamples to their conjecture.

math.CO↗

arXiv2026

Permanental Energy of Graphs

For a simple graph $G$ with adjacency matrix $A(G)$, let $π(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $μ_1,\ldots,μ_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n |μ_i|$. We prove a sharp universal lower bound: for every $m$-edge graph $G$, $E_{\mathrm{per}}(G) \ge 2\sqrt{m}$, with equality if and only if $G$ is a star together with isolated vertices. We also prove the general upper bound $E_{\mathrm{per}}(G) \le nρ(G)$, where $ρ(G)$ is the spectral radius, and we study $E_{\mathrm{per}}(G)$ on several graph families.

math.CO↗

arXiv2026

On Chollet's Permanent Conjecture for Graph Laplacians

In 1982, Chollet conjectured that $\mathrm{per}(A\circ B)\le \mathrm{per}(A)\mathrm{per}(B)$ for Hermitian positive semidefinite matrices $A,B$, where $\circ$ denotes the Hadamard product, and observed that in the real symmetric case it suffices to prove $\mathrm{per}(A\circ A)\le \mathrm{per}(A)^2$. We prove $\mathrm{per}(A\circ A)\le \mathrm{per}(A)^2$ for symmetric $Z$-matrices with nonnegative diagonal whose support graph is bipartite. Motivated by this, we study the Laplacian inequality $\mathrm{per}(L_G\circ L_G)\le \mathrm{per}(L_G)^2$ for the graph Laplacian $L_G$. We introduce a compositional framework for permanental inequalities on graph Laplacians, showing that Chollet's inequality is preserved under vertex coalescence. This enables the extension of the inequality from basic graph classes to large structured families, revealing new tractable regimes for a fundamentally $\#P$-hard quantity.

math.CO↗

arXiv2025

Permanental Analog of the Rank-Nullity Theorem for Symmetric Matrices

The rank of an n x n matrix A is equal to the size of its largest square submatrix with a nonzero determinant, and it can be computed in O(n^2.37) time. Analogously, the size of the largest square submatrix with nonzero permanent is defined as the permanental rank. Computing the permanent or the coefficients of the permanental polynomial is #P-complete. The permanental nullity is defined as the multiplicity of zero as a root of the permanental polynomial. We establish a permanental analog of the rank-nullity theorem, showing that the sum of the permanental rank and the permanental nullity equals n for symmetric nonnegative matrices, positive semidefinite matrices, and adjacency matrices of balanced signed graphs. Using this theorem, we can compute the permanental nullity for symmetric nonnegative matrices and adjacency matrices of balanced signed graphs in polynomial time. For symmetric matrices with entries in {0, plus or minus 1}, we also provide a complete characterization of when the permanental rank-nullity identity holds.

math.CO↗