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The purpose of this paper was to introduce the concepts of very thin multigroup, nondistributive (very thin) multirings, zero-divisor elements of multirings and zero-divisor graphs based on zero-divisor elements of multirings. In order to realize the article’s goals, we consider the relationship between finite nondistributive (very thin) multirings and multirings and construct nondistributive (very thin) multirings based on given ring. By some conditions on prime numbers, finite (nondistributive) very thin multirings are constructed. Zero-divisor graph based on zero-divisor set of (nondistributive) (very thin) multirings are introduced, so we investigate of some necessity and sufficiency conditions such that compute of order and size of these zero-divisor graphs. Also, the notations of derivable zero-divisor graphs and derivable zero-divisor subgraphs are introduced and is showed that some multipartite graphs are derivable zero-divisor graphs, all complete graphs, and cyclic graphs are derivable zero-divisor subgraphs.

A 2-edge-coloured graph G is supereulerian if G contains a spanning closed trail in which the edges alternate in colours. We show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. An eulerian factor of a 2-edge-coloured graph is a collection of vertex-disjoint induced subgraphs which cover all the vertices of G such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test whether a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2-edge-coloured graph is (trail-)colour-connected if it contains a pair of alternating (u, v)-paths ((u, v)-trails) whose union is an alternating closed walk for every pair of distinct vertices u, v. We prove that a 2-edge-coloured complete bipartite graph is supereulerian if and only if it is colour-connected and has an eulerian factor. A 2-edge-coloured graph is M-closed if xz is an edge of G whenever some vertex u is joined to both x and z by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in Contreras-Balbuena et al. (Discret Math Theoret Comput Sci 21:1, 2019), form a rich generalization of 2-edge-coloured complete graphs. We show that if G is obtained from an M-closed 2-edge-coloured graph H by substituting independent sets for the vertices of H, then G is supereulerian if and only if G is trail-colour-connected and has an eulerian factor. Finally we discuss 2-edge-coloured complete multipartite graphs and show that such a graph may not be supereulerian even if it is trail-colour-connected and has an eulerian factor.

In his paper "Kings in Bipartite Hypertournaments" (Graphs $\&$ Combinatorics 35, 2019), Petrovic stated two conjectures on 4-kings in multipartite hypertournaments. We prove one of these conjectures and give counterexamples for the other.

It is well known that the quantum mutual information and its conditional version do not increase under local channels. I this paper we show that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual information under local channels considered as a function on the Cartesian product of the set of all states of a composite system and the sets of all local channels (equipped with the strong convergence). Some applications of this property are considered. New continuity conditions for the quantum mutual information and for the squashed entanglement in both bipartite and multipartite infinite-dimensional systems are obtained. It is proved, in particular, that the multipartite squashed entanglement of any countably-non-decomposable separable state with finite marginal entropies is equal to zero. Special continuity properties of the information gain of a quantum measurement with and without quantum side information are established that can be treated as robustness (stability) of these quantities w.r.t. perturbation of the measurement and the measured state., Comment: 32 pages, preliminary version, any comments and references are welcome

We present a new quantum-memory-assisted entropic uncertainty relation for multipartite systems which shows the uncertainty principle of quantum mechanics. Notably, our results recover some well-known entropic uncertainty relations for two arbitrary incompatible observables that demonstrate the uncertainties about the results of two measurements. This uncertainty relation might play a critical role in the foundations of quantum theory.

The Seidel energy of a simple graph G is the sum of the absolute values of the eigenvalues of the Seidel matrix of G. In this paper we study the Seidel eigenvalues of complete multipartite graphs and find the exact value of the Seidel energy of the complete multipartite graphs.

An edge-colored graph is called rainbow if all its edges are colored distinct. The anti-Ramsey number of a graph family $${\mathcal {F}}$$ in the graph G, denoted by $$AR{(G,{\mathcal {F}})}$$ , is the maximum number of colors in an edge-coloring of G without rainbow subgraph in $${\mathcal {F}}$$ . The anti-Ramsey number for the short cycle $$C_3$$ has been determined in a few graphs. Its anti-Ramsey number in the complete graph can be easily obtained from the lexical edge-coloring. Gorgol considered the problem in complete split graphs which contains complete graphs as a subclass. In this paper, we study the problem in the complete multipartite graph which further enlarges the family of complete split graphs. The anti-Ramsey numbers for $$C_3$$ and $$C_3^{+}$$ in complete multipartite graphs are determined. These results contain the known results for $$C_3$$ and $$C_3^{+}$$ in complete and complete split graphs as corollaries.

We provide a characterization of multipartite systems constraints in terms of quantum correlations. By using the Hamming weight of the binary vectors associated with the subsystems, we give the α th power of monogamy and β th power of polygamy inequalities for general quantum correlations. Using concurrence as an application, one gets tighter inequalities than the existing ones for some classes of quantum states. Detailed examples are presented.

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.

An interconnection network is a complex connection of a set of processors and communication links between different processors which is utilized to exchange data between processors in the parallel computing system. Graph embedding is a vital tool used to run parallel algorithms in computer networks in which placing different modules on the board is one of the main cost criteria. By finding the optimal layout, the placement problem in circuit designs which does not have any deterministic algorithms can be solved. The complete multipartite graph is a widely used network topology with a high degree of reliability due to its flexible choice of network size. Recently a study was made by computing the optimal layout of balanced complete multipartite graphs into host graphs taken from the Cartesian product of elements in the set {path, cycle}, but leaving the case of product of paths as an open problem. This paper aims to solve such an open problem and the key idea is to locate suitable optimal sets in balanced complete multipartite graphs with respect to maximizing the number of edges in the located sets. Besides, we compute the layout in the case of host graphs like k-rooted complete binary trees, k-rooted sibling trees and the Cartesian product of path P 2 and complete binary trees.

The skewness of a graph G, denoted by sk(G), is the minimum number of edges in G whose removal results in a planar graph. It is an important parameter that measures how close a graph is to planarity, and it is complementary, and computationally equivalent, to the Maximum Planar Subgraph Problem. For any connected graph G on p vertices and q edges with girth g, one can easily verify that sk(G) ≥ π(G), where $$\pi(G)=\lceil q-\frac{g}{g-2}(p-2)\rceil$$ , and the graph G is said to be π-skew if equality holds. The concept of π-skew was first proposed by G. L. Chia and C. L. Lee. The π-skew graphs with girth 3 are precisely the graphs that contain a triangulation as a spanning subgraph. The purpose of this paper is to explore the properties of π-skew graphs. Some families of π-skew graphs are obtained by applying these properties, including join of two graphs, complete multipartite graphs and Cartesian product of two graphs. We also discuss the threshold for the existence of a spanning triangulation. Among other results some sufficient conditions regarding the regularity and size of a graph, which ensure a spanning triangulation, are given.

We investigate the monogamy relations of multipartite entanglement in terms of the α th power of concurrence, entanglement of formation, negativity and Tsallis-q entanglement. Enhanced new monogamy relations of multipartite entanglement with tighter lower bounds than the existing monogamy relations are presented, together with detailed examples showing the tightness. These monogamy relations give rise to finer characterization of the entanglement distributions among the subsystems of a multipartite system.

We consider the Bell experiment for a system described by multipartite states in the case where n dichotomic observables are measured per site. If n is two, we consider a two-setting Bell experiment. If n is M, we consider a M-setting Bell experiment. Two-setting model is an explicit local realistic model for the values of a correlation function, given in a two-setting Bell experiment. M-setting model is an explicit local realistic model for the values of a correlation function, given in a M-setting Bell experiment. Surprisingly we can discuss incompleteness in the Bell theorem. Also, we show a loophole problem of all the Bell-CHSH experimental claims. We discuss every Bell-CHSH experiment admits a local realistic theory if we rule out probability and statistics from the analysis of the experiment. We cannot obtain a violation of a Bell-CHSH inequality when we use only number theory and we do not use probability and statistics.

For a simple connected graph G, the Q-generating function of the numbers $$N_k$$ of semi-edge walks of length k in G is defined by $$W_Q(t)=\sum \nolimits _{k = 0}^\infty {N_k t^k }$$ . This paper reveals that the Q-generating function $$W_Q(t)$$ may be expressed in terms of the Q-polynomials of the graph G and its complement $$\overline{G}$$ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function $$W_Q(t)$$ , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix $$(\lambda I_n-Q(G))^{-1}$$ . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs.

The spectrum of the normalized Laplacian matrix of a graph provides many structural information of the graph, and it has many applications in numerous areas and in different guises. Let G be a complete k -partite graph with k ≥ 3 . In this paper, we give the necessary and sufficient condition for G which is determined by their normalized Laplacian spectrum. Moreover, we obtain a majorization theory of normalized Laplacian spectral radius of G , which enables us to find the maximal and minimal normalized Laplacian spectral radii among all complete k -partite graphs with fixed order, respectively.

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for \emph{metric graphs} by constructing for any positive rank divisor on a metric graph $\Gamma$ a positive rank divisor of the same degree on a subdivision of the underlying graph. Finally, we show that the treewidth lower bound also holds for a related notion of gonality defined by Caporaso and for stable gonality as introduced by Cornelissen et al., Comment: Changes w.r.t. v1: Expanded section on metric graphs, minor revisions in exposition, corrected small mistakes in proof

We find the Colin de Verdiere number μ for all complete multipartite graphs and show that for any graph G there exists a reduction process that results in a complete multipartite graph H with μ ( G ) ≤ μ ( H ) . We give examples of the upper bounds that can be obtained via this process.

Let be an integer. A graph is called r – partite if V admits a partition into r parts such that every edge has its ends in different parts. All of the r – partite graphs with given integer r consist of the class of multipartite graphs. Let be the set of multipartite graphs with r vertex parts, n nodes and diameter D. In this paper, we characterize the graphs with the maximum spectral radius in Furthermore, we show that the maximum spectral radius is not only a decreasing function on D, but also an increasing function on r.

Graph embedding is an important technique that maps a guest graph into a host graph, usually an interconnection network. Many applications can be modeled as graph embedding. In architecture simulation, graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. The quality of an embedding can be measured by certain cost criteria. One of these criteria is the wirelength and has been well studied by many authors (Lai and Tsai, 2010 [ 18 ]; Fan and Jia, 2007 [ 10 ]; Han et al., 2010 [ 15 ]; Fang and Lai, 2005 [ 11 ]; Park and Chwa, 2000 [ 23 ]; Rajasingh et al., 2004; Yang et al., 2010; Yang, 2009 [ 27 ]; Manuel et al., 2009; Bezrukov et al., 1998; Rostami and Habibi, 2008 [ 26 ]; Choudum and Nahdini, 2004 [ 7 ]; Guu, 1997 [ 14 ]). In this paper, we compute the exact wirelength of embedding complete multipartite graphs into certain graphs, such as path, cycle, wheel, hypertree, cylinder, torus and 3-regular circulant graphs.

In this paper, we present some new $Z$ -eigenvalue inclusion theorem for tensors by categorizing the entries of tensors, and prove that these sets are more precise than existing results. On this basis, some lower and upper bounds for the $Z$ -spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. As applications, we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors, and our results are more precise than existing result in some situations. In particular, for a given symmetric multipartite pure state with nonnegative amplitudes in real field, some theoretical lower and upper bounds for the geometric measure of entanglement are also derived in terms of the bounds for $Z$ -spectral radius. Numerical examples are given to illustrate validity and superiority of our results.

For simple graphs G and H the size multipartite Ramsey number mj ( G , H ) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m 3( G , H ) where G=mK1,n and H=P3 . In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3 . Moreover, we also determine m2(G,H) for this pair of graphs G and H .

The notions of critical ideals and characteristic ideals of graphs are introduced by Corrales and Valencia to study properties of graphs, including clique number, zero forcing number, minimum rank and critical group. In this paper, we provide methods to compute critical ideals of complete multipartite graphs and obtain complete answers for the characteristic ideals of complete multipartite graphs.

A signed graph $\Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $\Gamma(G)$. The energy of a signed graph $\Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(\Gamma(G))$ of $\Gamma(G)$. The random signed graph model $\mathcal{G}_n(p, q)$ is defined as follows: Let $p, q \ge 0$ be fixed, $0 \le p+q \le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs., Comment: 11 page, 0 figures. arXiv admin note: text overlap with arXiv:0909.4923 by other authors

Let K p 1 , p 2 , … , p r be a complete r-partite graph with r ≥ 2 and p i ≥ 2 . Varghese et al. (2018) [14] conjectured that the distance energy of K p 1 , p 2 , … , p r is always increased when an edge is deleted. In this paper, we prove that it is true for K p , p , … , p and the tripartite Turan graph T ( n , 3 ) . Subsequently, it is proved that, for any positive integer r, there exists a class of graphs that have r positive eigenvalues, but the distance energy of these graphs is always increased when an edge is deleted.

The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erd\H{o}s and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order $m$ in a direct product of balanced complete multipartite graphs., Comment: 5 pages, 1 figure

One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the α th power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2 ⊗ 2 ⊗ 3 systems and 2 ⊗ 2 ⊗ 2 n systems for α ≥ 4 ln 2 . We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0 ≤ β ≤ 2 . Given that the logarithmic negativity and LCREN are not convex, these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, 2 ⊗ 2 ⊗ 3 systems and 2 ⊗ 2 ⊗ 2 n systems in terms of the α th power of logarithmic negativity and LCREN for α ≥ 4 ln 2 . We also show that the β th power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0 ≤ β ≤ 2 .

In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.

We investigate the entanglement and nonlocality dynamics of a two-qubit Bell state and the three-qubit Greenberger–Home–Zeilinger (GHZ) state in a noisy environment by solving analytically a master equation in the Lindblad form. In the case of the Bell state, we obtain the analytic expressions of the concurrence and the maximal violation of the Clauser–Horne–Shimony–Holt inequality of the evolution state. In the case of the three-qubit GHZ state, we obtain the analytic formula of the genuinely multipartite concurrence of the evolution state. Usually, it is difficult to give the analytic expression of the maximum violation of the Svetlichny inequality for a general three-qubit state, but we find the analytic expression for a class of three-qubit states whose correlation matrices are of a special form. We use this result to obtain the analytic expression of the maximum violation of the Svetlichny inequality of the evolution state when parameters in the master equation meet certain conditions.

Monogamy and polygamy relations characterize the distributions of entanglement in multipartite systems. We investigate the monogamy and polygamy relations with respect to any partitions for a superposition of the generalized $W$-class state and vacuum in terms of the Tsallis-$q$ entanglement and the R\'enyi-$\alpha$ entanglement. By using the Hamming weight of the binary vectors related to the partitions of the subsystems, new classes of monogamy and polygamy inequalities are derived, which are shown to be tighter than the existing ones. Detailed examples are presented to illustrate the finer characterization of entanglement distributions., Comment: 17 pages, 4 figures

We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability and the optimal trade-off between the type-I and type-II errors, thus establishing the associated Chernoff, Stein, Hoeffding, and the strong converse exponents. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB., Comments are welcome

In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalising Dupuis' seminal single sender decoupling theorem. We start off with a multipartite quantum state, say on $A_{1}A_{2}R$ , where $A_{1}, A_{2}$ are treated as the two ‘sender’ systems and $R$ is the reference system. We apply independent Haar random unitaries in tensor product on $A_{1}$ and $A_{2}$ and then send the resulting systems through a quantum channel. We want the channel output $B$ to be almost in tensor with the untouched reference $R$ . Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a $k$ -sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard et al. [20] in the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

For any given integer $r\geqslant 3$, let $k=k(n)$ be an integer with $r\leqslant k\leqslant n$. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. Let $A_1,\ldots,A_k$ be a given $k$-partition of $[n]$ with $|A_i|=n_i\geqslant 1$. An $r$-uniform hypergraph $H$ is called {\it $k$-partite} if each edge $e$ satisfies $|e\cap A_i|\leqslant 1$ for $1\leqslant i\leqslant k$. In this paper, the number of linear $k$-partite $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the number of edges is $m(n)=o(n^{\frac{4}{3}})$. For $k=n$, it is the number of linear $r$-uniform hypergraphs on vertex set $[n]$ with $m=o(n^{ \frac{4}{3}})$ edges., accepted by Graphs and Combinatorics. arXiv admin note: text overlap with arXiv:2007.12490

Characterizing the zero entries in multipartite unitary matrices plays an important role in evaluating the usefulness of such matrices in quantum information. We investigate the zero entries in unitary and product unitary matrices. On one hand, we study the quantity properties of the zero entries in the multiqubit and bipartite product unitary matrices, respectively. We also investigate the ratio of the number of elements in the set made up of the numbers of zero entries in product unitary matrices to the number of all cases of such entries. On the other hand, we focus on the distribution of zero entries in unitary and product unitary matrices. Our results help identify whether a matrix containing at least one zero entry is a unitary matrix.

Monogamy and polygamy relations characterize the distribution of entanglement for multipartite systems. In this paper, we establish a class of tighter weighted monogamy relations of multi-qubit entanglement based on the α th-power of Tsallis-q entanglement(TE) for α ≥ 1 and 2 ≤ q ≤ 3. And for 0 ≤ α ≤ 1 and 1 ≤ q ≤ 2 or 3 ≤ q ≤ 4, the α th-power of Tsallis-q entanglement of assistance(TEoA) follows a class of tighter weighted polygamy relations. For α

In this article, we shall investigate the numerical invariants encoded in the minimal graded free resolution of edge ideal of k-partite n-crown graph(or multipartite crown graph) Cn(k). We ...

We demonstrate that a quantum hypergraph state is k-separable if and only if the hypergraph has k-connected components. The permutation symmetric states remains invariant under any permutation. We introduce permutation symmetric states generated by hypergraphs and describe their combinatorial structures. This combinatorial perspective insists us to investigate multi-partite entanglement of permutation symmetric hypergraph states. Using generalised concurrence we measure entanglement up to ten qubits. A number of examples of these states are discussed.

The monogamy relations of quantum correlation restrict the sharability of quantum correlations in multipartite quantum states. We show that all measures of quantum correlations satisfy some kind of monogamy relations for arbitrary multipartite quantum states. Moreover, by introducing residual quantum correlations, we present tighter monogamy inequalities that are better than all the existing ones. In particular, for multi-qubit pure states, we also establish new monogamous relations based on the concurrence and concurrence of assistance under the partition of the first two qubits and the remaining ones., 12 pages, 3 figures

In this paper we investigate bounds on set multipartite Ramsey numbers for the bipartite graph K 2 , n , extending or improving well-known upper bounds by Chung and Graham, Irving, Lortz and Mengersen. Known constructions based on certain classes of combinatorial designs (projective plane, Hadamard matrix, strongly regular graph) yield near-optimal bounds. As the main goal, a new construction based on strongly regular graph and Hadamard matrix produces a sharp class, generalizing a classical bound by Exoo, Harborth, and Mengersen.

Suppose the edges of K n are assigned weights by a weight function w . We define the weighted extremal number ex ( n , w , F ) ≔ max { w ( G ) ∣ G ⊆ K n , and G is F -free } where w ( G ) ≔ ∑ e ∈ E ( G ) w ( e ) . In this paper we study this problem for two types of weights w , each of which has an application. The first application is to an extremal problem in a complete multipartite host graph. The second application is to the maximum rectilinear crossing number of trees of diameter 4.

A bi-block graph is a connected graph all of whose blocks are complete bipartite graphs. We give a combinatorial interpretation of the Moore–Penrose inverse of the incidence matrix of a complete multipartite graph and a bi-block graph.

We present the results of our study of correlations in noncomposite systems like qudits (N-level atom or spin-j system). We show that they correspond to correlations in composite systems like bipartite or multipartite systems. We establish the correspondence between correlations in noncomposite systems and composite ones using the map of indices labeling the random variables in classical probability theory or matrix elements of density matrices in quantum mechanics. We obtain new information-entropic inequalities for density matrices of noncomposite systems. Also we present Bayes’ formula for noncomposite classical system with one random variable. Finally we discuss the possibility to use hidden correlations and entanglement of a single-qudit state.

We investigate genuine multipartite entanglement in general multipartite systems. Based on the norms of the correlation tensors of a multipartite state under various partitions, we present an analytical sufficient criterion for detecting the genuine four-partite entanglement. The results are generalized to arbitrary multipartite systems.

Multipartite secret sharing schemes have been an important object of study in the area of secret sharing schemes. Two interesting families of multipartite access structures are hierarchical access structures and compartmented access structures. This work deals with efficient and explicit constructions of ideal compartmented secret sharing schemes, while most of the known constructions are either inefficient or randomized. We construct ideal linear secret sharing schemes for three types of compartmented access structures, such as compartmented access structures with upper bounds, compartmented access structures with lower bounds, and compartmented access structures with upper and lower bounds. There exist some methods to construct ideal linear schemes realizing these compartmented access structures in the literature, but those methods are inefficient in general because non-singularity of many matrices has to be determined to check the correctness of the scheme. Our constructions do not need to do these computations. Our methods to construct ideal linear schemes realizing these access structures combine polymatroid-based techniques with Gabidulin codes. Gabidulin codes play a fundamental role in the constructions, and their properties imply that our methods are efficient.

The Kochen-Specker theorem is investigated for n spin-1/2 systems by using an inequality proposed in Nagata (J. Math. Phys. 46, 102101, 2005) on the basis on binary logic. A measurement theory based on the truth values (binary logic), i.e., the truth T (1) for true and the falsity F (0) for false is used. The values of measurement outcome are either + 1 or 0 (in $\hbar /2$ unit). The quantum predictions by n-multipartite states violate the inequality by an amount that grows exponentially with n. The measurement theory based on the binary logic provides an exponentially stronger refutation of the existence of hidden-variable when the number of parties of the state increases more. It turns out that the Kochen-Specker theorem becomes a quite strong theorem when the dimension of the multipartite state highly increases, regardless of entanglement properties.

Let D be a digraph. The competition graph of D is the graph sharing the same vertex set with D such that two different vertices are adjacent if and only if they have a common out-neighbor in D; the phylogeny graph of D is the competition graph of the digraph obtained from D by adding a loop at every vertex. For any graph G with n vertices, its competition number $$\kappa (G)$$ is the least nonnegative integer k such that G is an induced subgraph of the competition graph of an acyclic digraph with $$n+k$$ vertices, while its phylogeny number $$\phi (G)$$ is the least nonnegative integer p such that G is an induced subgraph of the phylogeny graph of an acyclic digraph with $$n+p$$ vertices. This paper provides new estimates of the competition numbers and phylogeny numbers of complete multipartite graphs with uniform part sizes. Accordingly, we can show that the range of the function $$\phi - \kappa +1$$ is the set of all nonnegative integers. We also report results about a hypergraph version of competition number and phylogeny number.

The Cohen–Macaulay property of a graph arising from a poset has been studied by various authors. In this article, we study the Cohen–Macaulay property of a graph arising from a family of reflexive ...

It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite graph, and if the different partition sets sizes are $p_1,\ldots, p_l$, and there are at least three partition sets of each size $p_i$, $i=1,\ldots, l$, then $G$ is determined, up to switching, by its Seidel spectrum. Sufficient conditions for a complete tripartite graph to be determined by its Seidel spectrum are discussed, and a conjecture is made on complete tripartite graphs on more than 18 vertices., 12 page 2 figures