Bipartite Chain Research Articles

Certifying Induced Subgraphs in Large Graphs

We introduce I/O-efficient certifying algorithms for the recognition of bipartite, split, threshold, bipartite chain, and trivially perfect graphs. When the input graph is a member of the respective class, the certifying algorithm returns a certificate that characterizes this class.Otherwise, it returns a forbidden induced subgraph as a certificate for non-membership.On a graph with $n$ vertices and $m$ edges, our algorithms take $\mathcal O(\text{sort}(n + m))$ I/Os in the worst case for split, threshold and trivially perfect graphs.In the same complexity bipartite and bipartite chain graphs can be certified with high probability.We provide implementations and an experimental evaluation for split and threshold graphs.

Topological magnon modes of a chain of magnetic spheres in a waveguide

We study topological features of a bipartite chain of magnetic spheres embedded in a rectangular metallic waveguide. The coherently coupled chain via the waveguide mode can host two topologically distinct phases identified by the Zak phase. A finite chain supports a pair of doubly degenerate topological edge states within the gap when the intercoupling dominates over the intracoupling, demonstrating that despite the strong coupling to the waveguide mode, the bulk-edge correspondence still holds. The nontrivial (trivial) topological phase is determined by the intracell and intercell coupling of magnetic spheres which depend on the separation, the external magnetic inductions, and the waveguide dimensions. Magnetically tunable topological magnon modes may enable unprecedented topological photonic devices.

Transitivity on subclasses of bipartite graphs

Let \(G=(V, E)\) be a graph where V and E are the vertex and edge sets, respectively. For two disjoint subsets A and B, we say A dominates B if every vertex of B is adjacent to at least one vertex of A. A vertex partition \(\pi = \{V_1, V_2, \ldots , V_k\}\) of G is called a transitive k-partition if \(V_i\) dominates \(V_j\) for all i, j where \(1\le i<j\le k\). The maximum integer k for which the above partition exists is called transitivity of G and it is denoted by Tr(G). The Maximum Transitivity Problem is to find a transitive partition of a given graph with the maximum number of parts. It was known that the decision version of Maximum Transitivity Problem is NP-complete for chordal graphs, which was proved by Hedetniemi et al. (Discret Math 278:81–108, 2004)]. This paper first strengthens the NP-completeness result by showing that this problem remains NP-complete for perfect elimination bipartite graphs. On the other hand, we propose a linear-time algorithm for finding the transitivity of a given bipartite chain graph. We then characterize graphs with transitivity at least t for any integer t. This result answers two open questions posed by J. T. Hedetniemi and S. T. Hedetniemi (J Combin Math Combin Comput 104:75–91, 2018).

Role coloring bipartite graphs

A k -role coloring α of a graph G is an assignment of k colors to the vertices of G such that every color is used at least once and if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n -role coloring. While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k -role coloring is a notoriously hard problem for k ≥ 2 . In fact, it is known that k - Role coloring is NP -complete for k ≥ 2 on general graph class. There has been extensive research on the complexity of k - role coloring on various hereditary graph classes. Furthering this direction of research, we show that k - Role coloring is NP -complete on bipartite graphs for k ≥ 3 (while it is trivial for k = 2 ). We complement the hardness result by characterizing 3-role colorable bipartite chain graphs, leading to a polynomial time algorithm for 3- Role coloring for this class of graphs. We further show that 2- Role coloring is NP -complete for graphs that are d vertices or edges away from the class of bipartite graphs, even when d = 1 .

Non-Floquet engineering in periodically driven dissipative open quantum systems

Floquet engineering plays a key role in realizing novel dynamical topological states. The conventional Floquet engineering, however, only applies to time-periodic non-dissipative Hermitian systems, and for the open quantum systems, non-Hermitian processes usually occur. So far, it remains unclear how to characterize the topological phases of time-periodic open quantum systems via the frequency space Floquet Hamiltonian. Here, we propose the non-Floquet theory to solve the problem and illustrate it by a continuously time-periodic non-Hermitian bipartite chain. In non-Floquet theory, a temporal non-unitary transformation is exercised on the Floquet states, and the transformed Floquet spectrum restores the form of the Wannier–Stark ladder. Besides, we also show that different choices of the starting points of the driving period can result in different localization behavior, effects of which can reversely be utilized to design quantum detectors of phases in dissipative oscillating fields. Our methods are capable of describing topological features in dynamical open quantum systems with various driving types and can find its applications to construct new types of dynamical topological materials.

Some spectral inequalities for connected bipartite graphs with maximum [formula omitted]-index

Let G be a graph, whose adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017 Nikiforov (2017) merged the A- and Q-spectral theories to proposed the Aα matrix: Aα(G)=αD(G)+(1−α)A(G), α∈[0,1]. Its largest eigenvalue is called the Aα-index of G. In this paper, we focus on specifying the property of the Aα matrix. On the one hand, we show that among the set of connected bipartite graphs of fixed order and size, the extremal graphs with the maximum Aα-index are the double nested graphs, i.e., bipartite chain graphs. On the other hand, we establish a series of inequalities respecting the entries of the Perron vector of Aα(G) of double nested graphs. Then we obtain some lower and upper bounds on the Aα-index of double nested graphs using the eigenvector and matrix techniques. Finally, we provide some computational results to compare these bounds.

Bipartite graphs of small readability

We study a parameter of bipartite graphs called readability, introduced by Chikhi et al. (Discrete Applied Mathematics, 2016) and motivated by applications of overlap graphs in bioinformatics. The behavior of the parameter is poorly understood. The complexity of computing it is open and it is not known whether the decision version of the problem is in NP. The only known upper bound on the readability of a bipartite graph (following from a work of Braga and Meidanis, LATIN 2002) is exponential in the maximum degree of the graph.Graphs that arise in bioinformatics applications have low readability. In this paper, we focus on graph families with readability o(n), where n is the number of vertices. We show that the readability of n-vertex bipartite chain graphs is between Ω(log⁡n) and O(n). We give an efficiently testable characterization of bipartite graphs of readability at most 2 and completely determine the readability of grids, showing in particular that their readability never exceeds 3. As a consequence, we obtain a polynomial time algorithm to determine the readability of induced subgraphs of grids. One of the highlights of our techniques is the appearance of Euler's totient function in the analysis of the readability of bipartite chain graphs. We also develop a new technique for proving lower bounds on readability, which is applicable to dense graphs with a large number of distinct degrees.

Tropical paths in vertex-colored graphs

A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the initial graph. In this work we study the problem of finding tropical paths in vertex-colored graphs. There are two versions for this problem: the shortest tropical path problem (STPP), i.e., finding a tropical path with the minimum total weight, and the maximum tropical path problem (MTPP), i.e., finding a path with the maximum number of colors possible. We show that both versions of this problems are NP-hard for directed acyclic graphs, cactus graphs and interval graphs. Moreover, we also provide a fixed parameter algorithm for STPP in general graphs and several polynomial-time algorithms for MTPP in specific graphs, including bipartite chain graphs, threshold graphs, trees, block graphs, and proper interval graphs.

Frustrated quantum magnetism in the Kondo lattice on the zigzag ladder

The interplay between Kondo effect, indirect magnetic interaction and geometrical frustration is studied in the Kondo lattice on the one-dimensional zigzag ladder. Using the density-matrix renormalization group (DMRG), the ground state and various short- and long-range spin- and density-correlation functions are calculated for the model at half-filling as a function of the antiferromagnetic Kondo interaction down to $J=0.3t$ where $t$ is the nearest-neighbor hopping on the zigzag ladder. Geometrical frustration is shown to lead to at least two critical points: Starting from the strong-$J$ limit, where almost local Kondo screening dominates and where the system is a nonmagnetic Kondo insulator, antiferromagnetic correlations between nearest-neighbor and next-nearest-neighbor local spins become stronger and stronger, until at $J^{\rm dim}_{\rm c} \approx 0.89t$ frustration is alleviated by a spontaneous breaking of translational symmetry and a corresponding transition to a dimerized state. This is characterized by antiferromagnetic correlations along the legs and by alternating antiferro- and ferromagnetic correlations on the rungs of the ladder. A mechanism of partial Kondo screening that has been suggested for the Kondo lattice on the two-dimensional triangular lattice is not realized in the one-dimensional case. Furthermore, within the symmetry-broken dimerized state, there is a magnetic transition to a $90^{\circ}$ quantum spin spiral with quasi-long-range order at $J^{\rm mag}_{\rm c} \approx 0.84t$. The quantum-critical point is characterized by a closure of the spin gap (with decreasing $J$) and a divergence of the spin-correlation length and of the spin-structure factor $S(q)$ at wave vector $q=\pi/2$. This is opposed to the model on the one-dimensional bipartite chain, which is known to have a finite spin gap for all $J>0$ at half-filling.

A New Graph Construction of Unbounded Clique-width

We define permutation-partition graphs by replacing one part of a 2K2-free bipartite graph (a bipartite chain graph) by an induced linear forest. We show that this hereditary graph class is of of unbounded clique-width (with a new graph construction of large clique-width). We show that this graph class contains no minimal graph class of unbounded clique-width, and give a conjecture for a contained boundary class for this property.

Some new considerations about double nested graphs

In the set of all connected graphs with fixed order and size, the graphs with maximal index are nested split graphs, also called threshold graphs. It was recently (and independently) observed in Bell et al. (2008) [3] and Bhattacharya et al. (2008) [4] that double nested graphs, also called bipartite chain graphs, play the same role within class of bipartite graphs. In this paper we study some structural and spectral features of double nested graphs. In studying the spectrum of double nested graphs we rather consider some weighted nonnegative matrices (of significantly less order) which preserve all positive eigenvalues of former ones. Moreover, their inverse matrices appear to be tridiagonal. Using this fact we provide several new bounds on the index (largest eigenvalue) of double nested graphs, and also deduce some bounds on eigenvector components for the index. We conclude the paper by examining the questions related to main versus non-main eigenvalues.

Polynomial kernels for Proper Interval Completion and related problems

Given a graph G=(V,E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (V×V)∖E such that the graph H=(V,E∪F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research. This problem is known to be FPT (Kaplan, Tarjan and Shamir, FOCSʼ94), but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with O(k3) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem, admits a kernel with O(k2) vertices, completing a previous result of Guo (ISAACʼ07).

Quantum level engineering for Aharonov-Bohm caging in the presence of electron-electron interactions

Aharonov-Bohm (AB) cages are finite regions of a quantum network where single-particle eigenstates remain extremely localized for certain values of an applied magnetic field. Electron-electron interactions partially destroy the single-particle localization by the generation of extended many-particle states. We address the effects of on-site potentials on the AB localization phenomenon in a diamond-shaped quasi-one-dimensional bipartite chain, and show that field-induced AB cages can effectively persist in the presence of electron-electron interactions.

Ising and Heisenberg models on ferrimagneticAB2chains

We study the Ising and Heisenberg models on one-dimensional ferrimagnetic bipartite chains with the special AB2 unit-cell topology and experimental motivation in inorganic and organic magnetic polymers. The spin-1/2 AB2 Ising case is exactly solved in the presence of an external magnetic field. We also derive asymptotical low- and high-temperature limits of several thermodynamical quantities of the zero-field classical AB2 Heisenberg model. Further, the quantum spin-1/2 AB2 Heisenberg model in a field is studied using a mean-field approach.

Dirichlet Forms on Totally Disconnected Spaces and Bipartite Markov Chains

We use Dirichlet form methods to construct and analyze a general class of reversible Markov precesses with totally disconnected state space. We study in detail the special case of bipartite Markov chains. The latter processes have a state space consisting of an “interior” with a countable number of isolated points and a, typically uncountable, “boundary.” The equilibrium measure assigns all of its mass to the interior. When the chain is started at a state in the interior, it holds for an exponentially distributed amount of time and then jumps to the boundary. It then instantaneously re-enters the interior. There is a “local time on the boundary.” That is, the set of times the process is on the boundary is uncountable and coincides with the points of increase of a continuous additive functional. Certain processes with values in the space of trees and the space of vertices of a fixed tree provide natural examples of bipartite chains. Moreover, time-changing a bipartite chain by its local time on the boundary leads to interesting processes, including particular Levy processes on local fields (for example, the p-adic numbers) that have been considered elsewhere in the literature.