Computing the permanental polynomial of $4k$-intercyclic bipartite graphs (2411.14238v1)

Abstract: Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $\phi(G,x)=\det(xI-A(G))$ and the permanental polynomial $\pi(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For bipartite graphs, we define the modified characteristic polynomial, which is obtained by changing the signs of some of the coefficients of $\phi(G,x)$. For $4k$-intercyclic bipartite graphs, i.e., those for which the removal of any $4k$-cycle results in a $C_{4k}$-free graph, we provide an expression for $\pi(G,x)$ in terms of the modified characteristic polynomial of the graph and its subgraphs. Our approach is purely combinatorial in contrast to the Pfaffian orientation method found in the literature to compute the permanental polynomial.

• The paper derives a novel formula linking the permanental and modified characteristic polynomials for 4k-intercyclic bipartite graphs.
• It shows that for C4k-free graphs, the permanental polynomial equals the modified characteristic polynomial, simplifying computation.
• The analysis paves the way for efficient algorithm development and encourages deeper exploration of combinatorial graph invariants.

Summary of "Computing the Permanental Polynomial of $4k$-Intercyclic Bipartite Graphs"

The paper, authored by Ravindra B. Bapat, Ranveer Singh, and Hitesh Wankhede, provides a comprehensive examination of the computation of the permanental polynomial for a specific class of bipartite graphs known as $4k$-intercyclic graphs. The primary contribution is the derivation of the permanental polynomial in terms of the modified characteristic polynomial, shedding light on the interplay between these two graph invariants and facilitating potentially more efficient computations.

Key Concepts

• Graph Invariants: The authors discuss two fundamental graph invariants: the characteristic polynomial $\phi(G,x)$ and the permanental polynomial $\pi(G,x)$. Traditionally, the characteristic polynomial has been extensively studied, largely due to its computational tractability compared to the permanental polynomial, which is $\#$P-complete.
• $4k$-Intercyclic Bipartite Graphs: These are bipartite graphs where the removal of any $4k$-cycle results in a $C_{4k}$-free graph. The paper introduces a technique to compute the permanental polynomial for these graphs using a combinatorial method rather than the more common Pfaffian orientation method.
• Modified Characteristic Polynomial: The authors propose using a modified version of the characteristic polynomial, adjusting the signs of certain coefficients for bipartite graphs, to express the permanental polynomial.

Key Results

• The authors offer a formula for the permanental polynomial of $4k$-intercyclic bipartite graphs through their modified characteristic polynomial. The approach circumvents the limitations of Pfaffian orientations, extending applicability to graphs beyond those that avoid an even subdivision of $K_{2,3}$.
• A significant finding is that for $C_{4k}$-free bipartite graphs, the permanental polynomial directly equals the modified characteristic polynomial, simplifying the computational effort considerably.
• The paper also provides an in-depth exploration of the mathematical identities involved in these computations, establishing connections between permanental and characteristic spectrums, especially in relation to the spectrum of the graph when dealing with $C_{4k}$-free cases.

Implications and Future Directions

From a practical standpoint, these findings may lead to more efficient algorithms for computing graph polynomials in specific cases, particularly when cycle structures are bounded or categorized by specific graph properties.

In terms of theoretical implications, this work invites further exploration into the class of $4k$-intercyclic graphs, potentially elucidating new characteristics or simplifying assumptions that can be generalized to broader classes of graphs. Moreover, these results may stimulate further research into other combinatorial methods for calculating otherwise intractable graph invariants.

Looking forward, future research could explore algorithmic implementations and optimizations leveraging the insights from this work. Additionally, the exploration of parallels with other graph-theoretic problems, such as graph isomorphism, and the extension of this paper's principles to non-bipartite graphs or larger classes of cycle-interaction-inferring graphs could offer promising fields of inquiry.

In conclusion, the paper significantly advances the understanding of permanental polynomials in $4k$-intercyclic bipartite graphs, offering a novel analytical lens and paving the way for future computational and theoretical advancements in combinatorics and graph theory.