A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

Evaluating EYM amplitudes in four dimensions by refined graphic expansion

The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − or $$ \left({h}_i^{-},{g}_j^{-}\right) $$ h i − g j − configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

Exponential growth constants for spanning forests on Archimedean lattices: Values and comparisons of upper bounds

We compare our upper bounds on the exponential growth constant [Formula: see text] characterizing the asymptotic behavior of spanning forests on Archimedean lattices [Formula: see text] with recently derived upper bounds.Our upper bounds on [Formula: see text], which are very close to the respective values of [Formula: see text] that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.

Extremal Trees with Fixed Degree Sequence

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

Functional Correctness of C Implementations of Dijkstra’s, Kruskal’s, and Prim’s Algorithms

We develop machine-checked verifications of the full functional correctness of C implementations of the eponymous graph algorithms of Dijkstra, Kruskal, and Prim. We extend Wang et al.’s CertiGraph platform to reason about labels on edges, undirected graphs, and common spatial representations of edge-labeled graphs such as adjacency matrices and edge lists. We certify binary heaps, including Floyd’s bottom-up heap construction, heapsort, and increase/decrease priority.Our verifications uncover subtle overflows implicit in standard textbook code, including a nontrivial bound on edge weights necessary to execute Dijkstra’s algorithm; we show that the intuitive guess fails and provide a workable refinement. We observe that the common notion that Prim’s algorithm requires a connected graph is wrong: we verify that a standard textbook implementation of Prim’s algorithm can compute minimum spanning forests without finding components first. Our verification of Kruskal’s algorithm reasons about two graphs simultaneously: the undirected graph undergoing MSF construction, and the directed graph representing the forest inside union-find. Our binary heap verification exposes precise bounds for the heap to operate correctly, avoids a subtle overflow error, and shows how to recycle keys to avoid overflow.

A Symmetric Function of Increasing Forests

For an indifference graph G, we define a symmetric function of increasing spanning forests of G. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular $\textrm {LLT}$ polynomials. As a consequence, we give a combinatorial interpretation of the coefficients of the $\textrm {LLT}$ polynomial in the elementary basis (up to a factor of a power of $(q-1)$ ), strengthening the description given in [4].