Priyanshu Pant

Priyanshu Pant, Ranveer Singh

Galluccio--Loebl and Tesler showed that the perfect-matching polynomial of a graph embedded in an orientable surface of genus $g$ can be written as a linear combination of at most $4^g$ Pfaffians. We show that, in general, exponentially many Pfaffians are necessary. More precisely, among all graphs of orientable genus at most $g$, the maximum possible Pfaffian number is at least $(8/3)^g$. This lower bound holds even for connected matching-covered graphs. We also obtain exponential lower bounds for the Pfaffian number of complete bipartite graphs, and hence for even complete graphs, improving asymptotically on a recent linear lower bound of Junchaya, Lucchesi, and Miranda.

Priyanshu Pant

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.

Priyanshu Pant, Ranveer Singh

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.

Priyanshu Pant, Ranveer Singh

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.