Genus Polynomials Research Articles

Partial-dual genus polynomial as a weight system

We prove that the partial-dual genus polynomial considered as a function on chord diagrams satisfies the four-term relation. Thus it is a weight system from the theory of Vassiliev knot invariants.

Partial-dual Euler-genus distributions for bouquets with small Euler genus

For an arbitrary ribbon graph G, the partial-dual Euler-genus polynomial of G is a generating function that enumerates partial duals of G by Euler genus. When G is an orientable ribbon graph, the partial-dual orientable genus polynomial of G is a generating function that enumerates partial duals of G by orientable genus. Gross, Mansour, and Tucker inaugurated these partial-dual Euler-genus and orientable genus distribution problems in 2020. A bouquet is a one-vertex ribbon graph. Given a ribbon graph G, its partial-dual Euler-genus polynomial is the same as that of some bouquet; this motivates our focus on bouquets. We obtain the partial-dual Euler-genus polynomials for all the bouquets with Euler genus at most two.

Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an analogous conjecture: The partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In this paper, we first give some counterexamples to the conjecture. Then motivated by these counterexamples, we further find two infinite classes of counterexamples.

Parallel edges in ribbon graphs and interpolating behavior of partial-duality polynomials

Recently, Gross, Mansour and Tucker introduced the partial-twuality polynomials. In this paper, we find that when there are enough parallel edges, any multiple graph is a negative answer to the problem 8.7 in their paper (Gross et al., 2021): Is the restricted-orientable partial-Petrial polynomial of an arbitrary ribbon graph even-interpolating? In addition, we also find a counterexample to the conjecture 8.1 of Gross, Mansour and Tucker: If the partial-dual genus polynomial is neither an odd nor an even polynomial, then it is interpolating.

Limit for the Euler-genus distributions of ladder-like sequences of graphs

In this paper, we demonstrate a method for calculating the production matrices for the Euler-genus polynomials of H-linear families of graphs. Particularly, for any ladder-like sequence of graphs, we give its explicit production matrix that leads to a recurrence relation. Based on this recurrence relation, we show that the Euler-genus distributions of any ladder-like sequence of graphs are asymptotic to a normal distribution. Since the genus polynomials of the ladder-like sequence of graphs have already been calculated by Chen et al.(J Algebr Combin 52:137–155, 2020), their crosscap-number polynomials are also known.

Counterexamples to a conjecture by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research problems and made some conjectures. In this paper, we introduce the notion of signed interlace sequences of bouquets and obtain the partial-dual Euler genus polynomials for all ribbon graphs with the number of edges less than 4 and the partial-dual orientable genus polynomials for all orientable ribbon graphs with the number of edges less than 5 in terms of signed interlace sequences. We check all the conjectures and find a counterexample to the Conjecture 3.1 in their paper: There is no orientable ribbon graph having a non-constant partial-dual genus polynomial with only one non-zero coefficient. Motivated by this counterexample, we further find an infinite family of counterexamples to the conjecture.

Recurrences for the genus polynomials of linear sequences of graphs

Abstract Given a finite graph H, the n th member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G n−1 by adding edges or identifying vertices, always in the same way. The genus polynomial Γ G (z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ Gn (z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, …, k with polynomials Γ G n j $\begin{array}{} \Gamma_{G_n}^j \end{array}$ (z), for j = 1, 2, …, k; from these polynomials, we form a column vector V n ( z ) = [ Γ G n 1 ( z ) , Γ G n 2 ( z ) , … , Γ G n k ( z ) ] t $\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ that satisfies a recursion Vn (z) = M(z)V n−1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a k th degree linear recursion for Γ n (z), allowing us to avoid the partitioning, thereby yielding a reduction from k 2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

Genus and Species-Level Classification of Wrasse Fishes Using Multidomain Features and Extreme Learning Machine Classifier

Automated recognition and classification of fishes are useful for studies dealing with counting of fishes for population assessments, discovering association between fishes and ecosystem, and monitoring of the ecosystem. This paper proposes a model which classifies the fishes belonging to the family Labridae in the genus and the species level. Features computed in the spatial and frequency domains are used in this work. All the images are preprocessed before feature extraction. Preprocessing step involves image segmentation for background elimination, de-noising and image enhancement. A combination of color, local binary pattern (LBP), histogram of oriented gradients (HOG), and wavelet features forms the feature vector. An ensemble feature reduction technique is used to reduce the attribute size. Performances of the system using combined as well as reduced feature sets are evaluated using seven popular classifiers. Among the classifiers, wavelet kernel extreme learning machine (ELM) showed higher classification accuracy of 96.65% in genus level and polynomial kernel ELM showed an accuracy of 92.42% in species level with the reduced feature set.

Partial duality for ribbon graphs, I: Distributions

The partial dualGA with respect to a subset A of edges of a ribbon graph G was introduced by Chmutov in connection with the Jones–Kauffman and Bollobás–Riordan polynomials, and it has developed into a topic of independent interest. This paper studies, for a given G, the enumeration of the partial duals of G by Euler genus, as represented by its generating function, which we call the partial-dual Euler-genus polynomial of G. A recursion is given for subdivision of an edge and is used to derive closed formulas for the partial-dual genus polynomials of four families of ribbon graphs. The log-concavity of these polynomials is studied in some detail. We include a concise, self-contained proof that χ(GA)=χ(A)+χ(E(G)−A)−2|V(G)|, where χ(G)=|V(G)|−|E(G)|+|F(G)|, and where A represents the ribbon graph obtained from G by deleting all edges not in A. This formula is a variant of a result of Moffatt.

Genus polynomials of ladder-like sequences of graphs

Production matrices have become established as a general paradigm for calculating the genus polynomials for linear sequences of graphs. Here we derive a formula for the production matrix of any of the linear sequences of graphs that we call ladder-like, where any connected graph H with two 1-valent root vertices may serve as a super-rung throughout the ladder. Our main theorem expresses the production matrix for any ladder-like sequence as a linear combination of two fixed \(3\times 3\) matrices, taken over the ring of polynomials with integer coefficients. This leads to a formula for the genus polynomials of the graphs in the ladder-like sequence, based on the two partial genus polynomials of the super-rung. We give a closed formula for these genus polynomials, for the case in which all imbeddings of the super-rung H are planar. We show that when the super-rung H has Betti number at most one, all the genus polynomials in the sequence are log-concave.

Genus polynomials and crosscap‐number polynomials for ring‐like graphs

AbstractAn H‐linear graph is obtained by transforming a collection of copies of a fixed graph H into a chain. An H‐ring‐like graph is formed by binding the two end‐copies of H in such a chain to each other. Genus polynomials have been calculated for bindings of several kinds. In this paper, we substantially generalize the rules for constructing sequences of H‐ring‐like graphs from sequences of H‐linear graphs, and we give a general method for obtaining a recursion for the genus polynomials of the graphs in a sequence of ring‐like graphs. We use Chebyshev polynomials to obtain explicit formulas for the genus polynomials of several such sequences. We also give methods for obtaining recursions for partial genus polynomials and for crosscap‐number polynomials of a bar‐ring of a sequence of disjoint graphs.

Calculating genus polynomials via string operations and matrices

To calculate the genus polynomials for a recursively specifiable sequence of graphs, the set of cellular imbeddings in oriented surfaces for each of the graphs is usually partitioned into imbedding-types . The effects of a recursively applied graph operation τ on each imbedding-type are represented by a production matrix . When the operation τ amounts to constructing the next member of the sequence by attaching a copy of a fixed graph H to the previous member, Stahl called the resulting sequence of graphs an H -linear family . We demonstrate herein how representing the imbedding types by strings and the operation τ by string operations enables us to automate the calculation of the production matrices, a task requiring time proportional to the square of the number of imbedding-types.

An Euler‐genus approach to the calculation of the crosscap‐number polynomial

AbstractIn 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap‐number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler‐genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap‐number polynomial for the nonorientable embeddings is then easily calculated from the Euler‐genus polynomial and the genus polynomial.

Fano plane’s embeddings on compact orientable surfaces

In this paper we study embeddings of the Fano plane as a bipartite graph. We classify all possible embeddings especially focusing on those with non-trivial automorphism group. We study them in terms of rotation systems, isomorphism classes and chirality. We construct quotients and show how to obtain information about face structure and genus of the covering embedding. As a by-product of the classification we determine the genus polynomial of the Fano plane.

Convolution preserves partial synchronicity of log-concave sequences

In a recent proof of the log-concavity of genus polynomials of some families of graphs, Gross et al. defined the weakly synchronicity relation between log-concave sequences, and conjectured that the convolution operation by any log-concave sequence preserves weakly synchronicity. We disprove it by providing a counterexample. Furthermore, we find the so-called partial synchronicity relation between log-concave sequences, which is (i) weaker than the synchronicity, (ii) stronger than the weakly synchronicity, and (iii) preserved by the convolution operation.

Iterated claws have real-rooted genus polynomials

We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly observable condition on their partitioned genus distributions, even though it had been proved previously that iterative amalgamation does not always preserve real-rootedness of the genus polynomial of the iterated graph. In this paper, the iterated topological operation is adding a claw, rather than vertex- or edge-amalgamation. Our analysis here illustrates some advantages of employing a matrix representation of the transposition of a set of productions.

Combinatorial conjectures that imply local log-concavity of graph genus polynomials

The 25-year old LCGD Conjecture is that the genus distribution of every graph is log-concave. We present herein a new topological conjecture, called the Local Log-Concavity Conjecture. We also present a purely combinatorial conjecture, which we prove to be equivalent to the Local Log-Concavity Conjecture. We use the equivalence to prove the Local Log-Concavity Conjecture for graphs of maximum degree four. We then show how a formula of David Jackson could be used to prove log-concavity for the genus distributions of various partial rotation systems, with straight-forward application to proving the local log-concavity of additional classes of graphs. We close with an additional conjecture, whose proof, along with proof of the Local Log-Concavity Conjecture, would affirm the LCGD Conjecture.

Log-Concavity of the Genus Polynomials of Ringel Ladders

A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder , that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polyno- mials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distribution of a symmetric ladder. Analysis of the resulting genus polynomial involves the use of Chebyshev polynomials. This paper continues our quest to affirm the quarter-century-old conjecture that all graphs have log-concave genus polynomials.

Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year-old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

Log-concavity of the genus polynomials for a sequence of cubic Halin graphs

Log-concavity of the genus polynomials for a sequence of cubic Halin graphs