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The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl. 609 (2021) 1-11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4. In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order n and rank r, and further determine the extremal graph which attains the minimum spectral radius among all graphs with order n and rank 5., Comment: 26 pages, 3 figures

Let D be a connected weighted digraph. The relation between the vertex- weighted complexity (with a fixed root) of the line digraph of D and the edge-weighted complexity (with a fixed root) of D ha...

Motivated by the connection with the genus of the corresponding link and its application on DNA polyhedral links, in this paper, we introduce a parameter smax(G), which is the maximum number of circles of states of the link diagram D(G) corresponding to a plane (positive) graph G. We show that smax(G) does not depend on the embedding of G and if G is a 4-edge-connected plane graph then smax(G) is equal to the number of faces of G, which cover the results of S. Y. Liu and H. P. Zhang as special cases.

Recently, Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph as a generating function that enumerates all partial duals of the ribbon graph by Euler genus. It is analogous to the extensively studied polynomial in topological graph theory that enumerates by Euler genus all embeddings of a given graph. To investigate the partial-dual polynomial, one only needs to focus on bouquets: that is, ribbon graphs with exactly one vertex. In this paper, we shall further show that the partial-dual polynomial of a bouquet essentially depends on the signed intersection graph of the bouquet rather than on the bouquet itself. That is to say, two bouquets with the same signed intersection graph have the same partial-dual polynomial. We then give a characterisation of when a bouquet has a planar partial dual in terms of its signed intersection graph. Finally, we consider a conjecture posed by Gross, Mansour and Tucker that there is no orientable ribbon graph whose partial-dual polynomial has only one nonconstant term; this conjecture is false, and we give a characterisation of when all partial duals of a bouquet have the same Euler genus.

Knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined and generalized them to virtual spatial graph theory in a combinatorial way and extended the Yamada polynomial from spatial graphs to virtual spatial graphs by ignoring virtual crossings. In this paper, we introduce a topological definition of the virtual spatial graph which is similar to that of a virtual link. Our main goal is to generalize the Yamada polynomial from spatial graphs to virtual spatial graphs such that it could capture the virtuality. It is realized via adding a new variable and integrating the topological Tutte polynomial instead of the Tutte polynomial into our polynomial. We define the generalized Yamada polynomial for virtual spatial graphs via their diagrams and prove that it can be normalized to be a rigid vertex isotopic invariant of virtual spatial graphs and to be a pliable vertex isotopic invariant for virtual spatial graphs with maximum degree at most 3. We consider the connection and difference between our generalized Yamada polynomial and the Dubrovnik polynomial of a classical link, and proved the generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for classical links which can be used to sometimes detect the virtuality of virtual links. We also prove the generalized Yamada polynomial specializes to the bracket polynomial of the associated diagram obtained from a virtual spatial graph diagram with the Jones–Wenzl projector P 2 acting on it, which can be used to write a program for calculating this special parametrization of the generalized Yamada polynomial based on Mathematica Code.

We study the computation of the Tutte polynomials of fan-like graphs and obtain expressions of their Tutte polynomials via generating functions. As applications, Tutte polynomials, in particular, the number of spanning trees, of two kinds of benzenoid systems, i.e. pyrene chains and triphenylene chains, are obtained.

Kotzig showed that every connected 4 -regular plane graph has an A -trail—an Eulerian circuit that turns either left or right at each vertex. However, this statement is not true for Eulerian plane graphs and determining if an Eulerian plane graph has an A -trail is NP-hard. The aim of this paper is to give a characterization of Eulerian embedded graphs having an A -trail. Andersen et al. showed the existence of orthogonal pairs of A -trails in checkerboard colourable 4 -regular graphs embedded on the plane, torus and projective plane. A problem posed in their paper is to characterize Eulerian embedded graphs (not necessarily checkerboard colourable) which contain two orthogonal A -trails. In this article, we solve this problem in terms of twisted duals. Several related results are also obtained.

In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings., 16 pages, 17 figures

In this paper, we first introduce the notions of checkerboard colourable minors for ribbon graphs motivated by the Eulerian ribbon graph minors, and two kinds of bipartite minors for ribbon graphs, one of which is the dual of the checkerboard colourable minors and the other is motivated by the bipartite minors of abstract graphs. Then we give an excluded minor characterization of the class of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.

Motivated by the connection with the genus of unoriented alternating links, Jin et al. (Acta Math Appl Sin Engl Ser, 2015) introduced the number of maximum state circles of a plane graph G, denoted by $$s_{\max }(G)$$ , and proved that $$s_{\max }(G)=\max \{e(H)+2c(H)-v(H)|$$ H is a spanning subgraph of $$G\}$$ , where e(H), c(H) and v(H) denote the size, the number of connected components and the order of H, respectively. In this paper, we show that for any (not necessarily planar) graph G, $$s_{\max }(G)$$ can be achieved by the spanning subgraph H of G whose each connected component is a maximal subgraph of G with two edge-disjoint spanning trees. Such a spanning subgraph is proved to be unique and we present a polynomial-time algorithm to find such a spanning subgraph for any graph G.

Self-similar graphs were widely studied in both combinatorics and statistical physics. Motivated by the construction of the well-known 3-dimensional Sierpinski gasket graphs, in this paper we introduce a family of recursively constructed self-similar graphs whose inner duals are of the self-similar property. By combining the dual property of the Tutte polynomial and the subgraph-decomposition trick, we show that the Tutte polynomial of this family of graphs can be computed in an iterative way and in particular the exact expression of the formula of the number of their spanning trees is derived. Furthermore, we show our method is a general one that is easily extended to compute Tutte polynomials for other families of self-similar graphs such as Farey graphs, 2-dimensional Sierpinski gasket graphs, Hanoi graphs, modified Koch graphs, Apollonian graphs, pseudofractal scale-free web, fractal scale-free network, etc.

In the paper Combinatorica 33(2) (2013) 231–252, Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. An open problem posed in their paper is the characterization of Eulerian partial duals of plane graphs. In this article, we solve this problem by considering half-edge orientations of medial graphs.

Let ?$G$? be a ribbon graph and ?$\mu (G)$? be the number of components of the virtual link formed from ?$G$? as a cellularly embedded graph via the medial construction. In this paper we first prove that ?$\mu (G) \leq f(G) + \gamma (G)$?, where ?$f(G)$? and ?$\gamma (G)$? are the number of boundary components and Euler genus of ?$G$?, respectively. A ribbon graph is said to be extremal if ?$\mu (G) = f(G) + \gamma (G)$?. We then obtain that a ribbon graph is extremal if and only if its Petrial is plane. We introduce a notion of extremal minor and provide an excluded extremal minor characterization for extremal ribbon graphs. We also point out that a related result in the monograph by J. A. Ellis-Monaghan and I. Moffatt Graphs on surfaces. Dualities, polynomials, and knots. New York, NY: Springer (2013) is not correct and prove that two related conjectures raised by S. Huggett and I. Tawfik Ars Math. Contemp. 8, No. 2, 319-335 (2015) hold for more general ribbon graphs. Naj bo ?$G$? trakovni graf in naj bo ?$\mu (G)$? število komponent virtualne povezave, tvorjene iz grafa ?$G$? kot celično vloženega grafa preko medialne konstrukcije. V članku najprej dokažemo, da je ?$\mu (G) \leq f(G) + \gamma (G)$?, kjer je ?$f(G)$? število robnih komponent, ?$\gamma (G)$? pa Eulerjev rod grafa ?$G$?. Trakovni graf je ekstremalen, če je ?$\mu (G) = f(G) + \gamma (G)$?. Nato pokažemo, da je trakovni graf ekstremalen, če in samo če je njegov Petrial ravninski graf. Vpeljemo pojem ekstremalnega minorja in podamo karakterizacijo ekstremalnih trakovnih grafov z izključenimi ekstremalnimi minorji. Izpostavimo tudi, da soroden rezultat v monografiji Ellis-Monaghana in Moffatta ni pravilen, ter dokažemo, da dve sorodni domnevi, ki sta ju postavila Huggett in Tawfik, veljata za splošnejše trakovne grafe.

The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. Twisted duality is a further generalization from combining partial Petrials with partial duals. Ribbon graphs are a new equivalent form of the old cellularly embedded graphs. In this paper, we first characterize regular partial duals of a ribbon graph by using spanning quasi-tree and its related nesting set. Then we characterize checkerboard colourable partial Petrials for any Eulerian ribbon graph by using spanning trees and a related notion of adjoint set. Finally we give a complete characterization of all regular checkerboard colourable twisted duals of a ribbon graph, which solves a problem raised by Ellis-Monaghan and Moffatt (2012) [10] .

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.

Benzenoid systems are natural graph representation of benzenoid hydrocarbons. Many chemically and combinatorially interesting indices and polynomials for bezenoid systems have been widely researched by both chemists and graph theorists. The Tutte polynomial of benzenoid chains without branched hexagons has already been computed by the recursive method. In this paper, by multiple recursion schema, an explicit expression for the Tutte polynomial of benzenoid systems with exactly one branched hexagon is obtained in terms of the number of hexagons on three linear or kinked chains. As a by-product, the number of spanning trees for these kind of benzenoid systems is determined.

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.

As projections of links, 4-regular plane graphs are important in combinatorial knot theory. The flow polynomial of 4-regular plane graphs has a close relation with the two-variable Kauffman polynomial of links. F. Jaeger in 1991 provided even subgraph expansions for the flow polynomial of cubic plane graphs. Starting from and based on Jaeger's work, by introducing splitting systems of even subgraphs, we extend Jaeger's results from cubic plane graphs to plane graphs with maximum degree at most 4 including 4-regular plane graphs as special cases. Several consequences are derived and further work is discussed.

Polyhedral links, interlinked and interlocked architectures, have been proposed for the description and characterization of DNA and protein polyhedra. In this paper, we study the topological chirality of a type of DNA polyhedral links constructed by the strategy of “n-point stars” and a type of protein polyhedral links constructed by “three-cross curves and untwisted double-line” covering. Furthermore, we prove that links corresponding to bipartite plane graphs have antiparallel orientations, and under these orientations, their writhes are not zero. As a result, the type of double crossover DNA polyhedral links are topologically chiral. We also prove that the unoriented link corresponding to a connected, even, bipartite plane graph always has self-writhe 0. Using the Jones polynomial for unoriented links we derive two simple criteria for chirality of unoriented alternating links with self-writhe 0. By applying this criterion we show that 3-regular protein polyhedral links are also topologically chiral. Topological chirality always implies chemical chirality, hence the corresponding DNA and protein polyhedra are all chemically chiral. Our chiral criteria may be used to detect the topological chirality of more complicated DNA and protein polyhedral links that may be synthesized by chemists and biologists in the future.

The partition function of q -state Potts model, or equivalently the Tutte polynomial, is computationally intractable for regular lattices. The purpose of this paper is to compute partition functions of q -state Potts model on two families of fractal lattices. Based on their self-similar structures and by applying the subgraph-decomposition method, we divide their Tutte polynomials into two summands, and for each summand we obtain a recursive formula involving the other summand. As a result, the number of spanning trees and their asymptotic growth constants, and a lower bound of the number of connected spanning subgraphs or acyclic root-connected orientations for each of such two lattices are obtained.

In this paper, we study connected plane graphs with link component number equal to the nullity and call them near-extremal graphs. We first study near-extremal graphs with minimum degree at least 3 and prove that a connected plane graph G with minimum degree at least 3 is a near-extremal graph if and only if G is isomorphic to K4, the complete graph with 4 vertices. The result is obtained by studying general graphs using the knowledge of bicycle space and the Tutte polynomial. Then a simple algorithm is given to judge whether a connected plane graph is a near-extremal graph or not. Finally we study the construction of near-extremal graphs and prove that all near-extremal graphs can be constructed from a loop and K4 by two graph operations.

National Natural Science Foundation of China [10831001]; Fundamental Research Funds for the Central Universities [2010121007]

In this paper, we first recall some known architectures of polyhedral links (Qiu and Zhai in J Mol Struct (THEOCHEM) 756:163–166, 2005; Yang and Qiu in MATCH Commun Math Comput Chem 58:635–646, 2007; Qiu et al. in Sci China Ser B Chem 51:13–18, 2008; Hu et al. in J Math Chem 46:592–603, 2009; Cheng et al. in MATCH Commun Math Comput Chem 62:115–130, 2009; Cheng et al. in MATCH Commun Math Comput Chem 63:115–130, 2010; Liu et al. in J Math Chem 48:439–456 2010). Motivated by these architectures we introduce the notions of polyhedral links based on edge covering, vertex covering, and mixed edge and vertex covering, which include all polyhedral links in Qiu and Zhai (J Mol Struct (THEOCHEM) 756:163–166, 2005), Yang and Qiu (MATCH Commun Math Comput Chem 58:635–646, 2007), Qiu et al. (Sci China Ser B Chem 51:13–18, 2008), Hu et al. (J Math Chem 46:592–603, 2009), Cheng et al. (MATCH Commun Math Comput Chem 62:115–130, 2009), Cheng et al. (MATCH Commun Math Comput Chem 63:115–130, 2010), Liu et al. (J Math Chem 48:439–456, 2010) as special cases. The analysis of chirality of polyhedral links is very important in stereochemistry and the Jones polynomial is powerful in differentiating the chirality (Flapan in When topology meets chemistry. Cambridge Univ. Press, Cambridge, 2000). Then we give a detailed account of a result on the computation of the Jones polynomial of polyhedral links based on edge covering developed by Jin, Zhang, Dong and Tay (Electron. J. Comb. 17(1): R94, 2010) and, at the same time, by using this method we obtain some new computational results on polyhedral links of rational type and uniform polyhedral links with small edge covering units. These new computational results are helpful to judge the chirality of polyhedral links based on edge covering. Finally, we give some remarks and pose some problems for further study.

For a signed cyclic graph [Formula: see text], we can construct a unique virtual link [Formula: see text] by taking the medial construction and converting 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In this paper, we shall prove that a virtual link [Formula: see text] is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial [Formula: see text] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between [Formula: see text] of a signed cyclic graph [Formula: see text] and the bracket polynomial of one of the virtual link diagrams associated with [Formula: see text]. Finally, we give a spanning subgraph expansion for [Formula: see text].

Let D be a general connected reduced alternating link diagram, C be the set of crossings of D and C′ be the nonempty subset of C. In this paper we first define a multiple crossing-twisted link family {D n (C′)|n=1,2,…} based on D and C′, which produces (2,2n+1)-torus knot family, the link family A n defined in Chang and Shrock (Physica A 301:196–218, 2001) and the pretzel link family P(n,n,n) as special cases. Then by applying Beraha-Kahane-Weiss’s Theorem we prove that limits of zeros of Jones polynomials of {D n (C′)|n=1,2,…} are the unit circle |z|=1 (It is independent of the selections of D and C′) and several isolated limits, which can be determined by computing flow polynomials of subgraphs of G corresponding to D. Furthermore, we use the method of Brown and Hickman (Discrete Math. 242:17–30, 2002) to prove that, for any e>0, all zeros of Jones polynomial of the link D n (C) lie inside the circle |z|=1+e, provided that n is large enough. Our results extend results of F.Y. Wu, J. Wang, S.-C. Chang, R. Shrock and the present authors and refine partial result of A. Champanerkar and L. Kofman.

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

It is well known that Kauffman constructed a state model of the Jones polynomial based on unoriented link diagrams. In his approach, in order to obtain Jones polynomial one needs to calculate both the writhe and the Kauffman bracket. Stimulated by a paper of Altintas (An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput.194 (2007) 168–178), in this paper we present a state sum model based on oriented link diagrams. In our approach, we succeed in adding the writhe to the state sum model and need not to compute the writher any more. We further show that, via our state sum model, Jones polynomial of any link (alternating or not) is a special parametrization of the dichromatic polynomial of a weighted graph with two different edge weights.

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K1 by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.

Based on the connection with Alexander polynomial of special alternating links, Murasugi and Stoimenow introduced the Alexander polynomial of even graphs. In this paper, we study the Alexander polynomial of spatial even graphs with reflective symmetry. Roughly speaking, we prove that the Alexander polynomial of one half of a spatial even graph with reflective symmetry is a divisor of that of the whole spatial even graph. Then, we apply the result to a family of special alternating links, expressing the Alexander polynomial of such a link as the product of Alexander polynomials of two smaller special alternating links derived from the two isotopic "halves" of the original link.

Let G be a signed graph. Let Gˆ be the graph obtained from G by replacing each edge e of G by a chain or a sheaf. In this paper we first establish a relation between the Q-polynomial of Gˆ [L.H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1989) 105–127] and the W-polynomial of G [L. Traldi, A dichromatic polynomial for weighted graphs and link diagrams, Proc. Amer. Math. Soc. 106 (1989) 279–286; B. Bollobás, O. Riordan, A Tutte polynomial for colored graphs, Combin. Probab. Comput. 8 (1999) 45–93]. Then we derive two special dual cases from the relation, one of which has been obtained in [X. Jin, F. Zhang, The Kauffman brackets for equivalence classes of links, Adv. in Appl. Math. 34 (2005) 47–64]. Based on the one-to-one correspondence between signed plane graphs and link diagrams, and the correspondence between the Q-polynomial of a signed plane graph and the Kauffman bracket of a link diagram, we can compute the Kauffman bracket of the link diagram corresponding to Gˆ by means of the W-polynomial of G. In this way we investigate the Kauffman brackets of rational links as a typical link family, and obtain an explicit formula using transfer matrix approach. Finally we provide another link family to point out that the relation we built actually can be used to deal with Kauffman brackets of all such link families.

In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.

In the last 20 years or so, chemists and molecular biologists have synthesized some novel DNA polyhedra. Polyhedral links were introduced to model DNA polyhedra and study topological properties of DNA polyhedra. As a very powerful invariant of oriented links, the Homfly polynomial of some of such polyhedral links with small number of crossings has been obtained. However, it is a challenge to compute Homfly polynomials of polyhedral links with large number of crossings such as double crossover 3-regular links considered here. In this paper, a general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions. As a result, we can obtain the Homfly polynomial of the double crossover 3-regular link which has relatively large number of crossings.

It is well known that there is a one-to-one correspondence between link diagrams and signed plane graphs. In this paper, we define an equivalence relation on the set of link diagrams according to the reductions of their corresponding signed plane graphs. For all members of each equivalence class (with infinite number of members), we can compute their Kauffman bracket polynomials in a unified way. As an example, we deal with the links whose reductions of their corresponding signed plane graphs have small cyclomatic numbers. Furthermore, applying computer algebra (MAPLE) techniques, we calculate the Kauffman bracket polynomials of a set of links in a specified equivalence class.

In this paper, we define a family of links which are similar to but more complex than Pretzel links. We compute the exact expressions of the Jones polynomials for this family of links. Motivated by the connection with the Potts model in statistical mechanics, we investigate accumulation points of zeros of the Jones polynomials for some subfamilies.

In this paper, a general method for computing the Tutte polynomial of the subdivision of a graph is explained. As an application to the subdivision of sheaf graph which consists of two vertices joined by some parallel edges, we obtain the explicit expressions of the Jones polynomials for some families of the pretzel links. Motivated by the work of Chang and Shrock, we investigate the zeros distribution of its Jones polynomial for each family when the number of crossings goes to infinity, and generalize some of their results.

Ichihara and Matsudo introduced the notions of [Formula: see text]-colorable links and the minimal coloring number for [Formula: see text]-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable [Formula: see text]-colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable [Formula: see text]-colorable link is exactly 4.

In this paper, we introduce a notion of ear decomposition of 3-regular polyhedral links based on the ear decomposition of the 3-regular polyhedral graphs. As a result, we obtain an upper bound for the braid index of 3-regular polyhedral links. Our results may be used to characterize and analyze the structure and complexity of protein polyhedra and entanglement in biopolymers.

In this paper, we present a formula for computing the Tutte polynomial of the signed graph formed from a labeled graph by edge replacements in terms of the chain polynomial of the labeled graph. Then we define a family of 'ring of tangles' links and consider zeros of their Jones polynomials. By applying the formula obtained, Beraha-Kahane-Weiss's theorem and Sokal's lemma, we prove that zeros of Jones polynomials of (pretzel) links are dense in the whole complex plane.

In this paper, we show that any 11-colorable knot has a diagram with a non-trivial 11-coloring such that at most six colors among eleven colors are assigned to arcs of the diagram.

In this paper, an expression for the Homfly polynomial of a classical pretzel link with a fixed orientation was obtained. As an application, we proved that a (p, q, r)-pretzel knot is achiral if and only if {p, q, r} = {2, 1, 1}, {-2, -1, -1}, {2, -1, 3}, {-2, 1, -3}, {1, -1, k}, {1, k, -2 -k} or {-1, -k, 2 + k}, where k is an odd integer.