Search

Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. Secret sharing schemes for multipartite access structures have received considerable attention due to the fact that multipartite secret sharing can be seen as a natural and useful generalization of threshold secret sharing.

Multipartite bacteria have one chromosome and one or more chromid. Chromids are believed to have properties that enhance genomic flexibility, making them a favored integration site for new genes. However, the mechanism by which chromosomes and chromids jointly contribute to this flexibility is not clear. To shed light on this, we analyzed the openness of chromosomes and chromids of the two bacteria, Vibrio and Pseudoalteromonas, both which belong to the Enterobacterales order of Gammaproteobacteria, and compared the genomic openness with that of monopartite genomes in the same order. We applied pangenome analysis, codon usage analysis and the HGTector software to detect horizontally transferred genes. Our findings suggest that the chromids of Vibrio and Pseudoalteromonas originated from two separate plasmid acquisition events. Bipartite genomes were found to be more open compared to monopartite. We found that the shell and cloud pangene categories drive the openness of bipartite genomes in Vibrio and Pseudoalteromonas. Based on this and our two recent studies, we propose a hypothesis that explains how chromids and the chromosome terminus region contribute to the genomic plasticity of bipartite genomes.

Interpersonal communication is so important in everyday life that it is desirable everyone who participates in the conversation is satisfied. However, every participant of the conversation cannot be satisfied in such cases as those wherein only one person cannot keep up with the conversation and feels alienated, or wherein someone cannot communicate non-verbal expressions with his/her conversation partner adequately. In this study, we have focused on facial direction and gaze among the various factors that are said to affect conversational satisfaction. We have attempted to lessen any sense of non-participation in the conversation and increase the conversational satisfaction of the non-participant in a tripartite conversation by modulating the visual information in such a way that the remaining two parties turn toward the non-participating party. In the experiments we have conducted in VR environments, we have reproduced a conversation of two male adults recorded in actual environments using two avatars. The experimental subjects have watched this over their HMDs. The experiments have found that visually modulating the avatars’ faces and gazes such that they appear to turn toward the subjects has increased the subjects’ sense of participation in the conversation. Nevertheless, the experiments have not increased the subjects’ conversational enjoyment, a component of the factors for conversational satisfaction.

We look into multipartite quantum states on which quantum cryptographic protocols including quantum key distribution and quantum secret sharing can be perfectly performed, and define the quantum cryptographic resource distillable rate as the asymptotic rate at which such multipartite state can be distilled from a given multipartite state. Investigating several relations between entanglement and the rate, we show that there exists a multipartite bound entangled state whose quantum cryptographic resource distillable rate is strictly positive, that is, there exists a multipartite entangled state which is not distillable, but can be useful for quantum cryptography such as quantum key distribution and quantum secret sharing., 9 pages, 2 figures

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.

Multipartite secret sharing schemes are those that have multipartite access structures. The set of the participants in those schemes is divided into several parts, and all the participants in the same part play the equivalent role. One type of such access structure is the compartmented access structure, and the other is the hierarchical access structure. We propose an efficient compartmented multisecret sharing scheme based on the linear homogeneous recurrence (LHR) relations. In the construction phase, the shared secrets are hidden in some terms of the linear homogeneous recurrence sequence. In the recovery phase, the shared secrets are obtained by solving those terms in which the shared secrets are hidden. When the global threshold is t , our scheme can reduce the computational complexity of the compartmented secret sharing schemes from the exponential time to polynomial time. The security of the proposed scheme is based on Shamir’s threshold scheme, i.e., our scheme is perfect and ideal. Moreover, it is efficient to share the multisecret and to change the shared secrets in the proposed scheme.

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.

In recent decades, multipartite mutualisms involving microorganisms such as fungi have been discovered in associations traditionally thought of as bipartite. Ant-plant mutualisms were long thought to be bipartite despite fungi being noticed in an epiphytic ant-plant over 100 years ago. We sequenced fungal DNA from the three distinct domatium chambers of the epiphytic ant-plant Myrmecodia beccarii to establish if fungal communities differ by chamber type across five geographic locations spanning 675 km. The three chamber types serve different ant-associated functions including ‘waste’ chambers, where ant workers deposit waste; ‘nursery’ chambers, where the brood is kept; and ‘ventilation’ chambers, that allow air into the domatium. Overall, fungi from the order Chaetothyriales dominated the chambers in terms of the proportion of operational taxonomic units (OTUs; 13.4%) and sequence abundances of OTUs (28% of the total); however a large portion of OTUs (28%) were unidentified at the order level. Notably, the fungal community in the waste chambers differed consistently from the nursery and ventilation chambers across all five locations. We identified 13 fungal OTUs as ‘common’ in the waste chambers that were rare or in very low sequence abundance in the other two chambers. Fungal communities in the nursery and ventilation chambers overlapped more than either did with the waste chambers but were also distinct from each other. Differences in dominance of the common OTUs drove the observed patterns in the fungal communities for each of the chamber types. This suggests a multipartite mutualism involving fungi exists in this ant-plant and that the role of fungi differs among chamber types.

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.

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.

Rhizobium leguminosarum symbiovar trifolii strains TA1 and CC275e are nitrogen-fixing microsymbionts of Trifolium spp. and have been used as commercial inoculant strains for clovers in pastoral agriculture in Australia and New Zealand. Here we present the complete genome sequences of both strains, resolving their multipartite genome structures and allowing for future studies using genomic approaches. [Formula: see text] Copyright © 2021 The Author(s). This is an open access article distributed under the CC BY 4.0 International license .

Pure quantum spin-s states can be represented by 2s points on the sphere, as shown by Majorana (Nuovo Cimento 9:43–50, 1932)—the description has proven particularly useful in the study of rotational symmetries of the states, and a host of other properties, as the points rotate rigidly on the sphere when the state undergoes an SU(2) transformation in Hilbert space. We present here an extension of this representation to multipartite, totally antisymmetric (under exchange of any two qudits) states, widely known in the form of Slater determinants, and linear combinations thereof. Such states generally involve a superposition of various spin values, giving rise to a family of Majorana-like constellations, that captures their rotational transformation properties. We also point out that our results apply equally well to the characterization of degenerate linear subspaces of the Hilbert space of a single spin, of the type that appear in the Wilczek–Zee effect, and comment on potential applications to holonomic quantum computing.

We calculate spin correlation functions using IBM quantum processors, accessed online. We demonstrate the rotational invariance of the singlet state, interesting properties of the triplet states, and surprising features of a state of three entangled qubits. This exercise is ideal for remote learning and generates data with real quantum mechanical systems that are impractical to investigate in the local laboratory. Students learn a wide variety of skills, including calculation of multipartite spin correlation functions, design and analysis of quantum circuits, and remote measurement with real quantum processors.

The human face is complex and multipartite, and characterization of its genetic architecture remains challenging. Using a multivariate genome-wide association study meta-analysis of 8,246 European individuals, we identified 203 genome-wide-significant signals (120 also study-wide significant) associated with normal-range facial variation. Follow-up analyses indicate that the regions surrounding these signals are enriched for enhancer activity in cranial neural crest cells and craniofacial tissues, several regions harbor multiple signals with associations to different facial phenotypes, and there is evidence for potential coordinated actions of variants. In summary, our analyses provide insights into the understanding of how complex morphological traits are shaped by both individual and coordinated genetic actions. ispartof: Nature Genetics vol:53 issue:1 ispartof: location:United States status: Published online

Synthetic biology relies on rapid and efficient methods to stably integrate DNA payloads encoding for synthetic biological systems into the genome of living cells. The size of designed biological systems increases with their complexity, and novel methods are needed that enable efficient and simultaneous integration of multiple payloads into single cells. By assembling natural and synthetic protein-protein dimerization domains, we have engineered a set of multipartite transcription factors for driving heterologous target gene expression. With the distribution of single parts of multipartite transcription factors on piggyback transposon-based donor plasmids, we have created a logic genome integration control (LOGIC) system that allows for efficient one-step selection of stable mammalian cell lines with up to three plasmids. LOGIC significantly enhances the efficiency of multiplexed payload integration in mammalian cells compared to traditional cotransfection and may advance cell line engineering in synthetic biology and biotechnology., ACS Synthetic Biology, 9 (11), ISSN:2161-5063

Certifying multipartite entanglement is a fundamental task. Since $n$ -qubit state is parameterized by $4^{n}-1$ real numbers, it is interesting to design a measurement setup that detects multipartite entanglement with as little effort as possible, and at a minimum without fully revealing the whole information of the state, the so-called “tomography”. In this paper, we study the relationship between multipartite entanglement certification and tomography, with the constraint that only single-copy measurements are allowed. We show that by using nonadaptive single-copy measurements, universal entanglement detection, among all states, can not be accomplished without full state tomography. Moreover, we show that almost all multipartite correlations, including the genuine entanglement and the entanglement depth, require full state tomography to detect in this measurement setting. We also observe that universal entanglement detection, among pure states, can be accomplished using much fewer measurements than full state tomography even using only local measurements.

Multipartite secret sharing is an important research object in the area of secret sharing schemes. Compartmented access structures are an interesting class of multipartite access structures. The constructions of ideal linear schemes realizing compartmented access structures are studied by codes in this paper. We find that compartmented secret sharing is related to a class of codes called locally repairable codes which are being used widely in distributed and cloud storage systems. We study secret sharing schemes for compartmented access structures with upper bounds, compartmented access structures with lowers bounds and compartmented access structures with upper and lower bounds. We establish the relationships between secret sharing schemes for these compartmented access structures and locally repairable codes. Based on the relationships and some locally repairable codes, we obtain ideal linear schemes realizing these compartmented access structures by efficient methods.

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed.

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

This article presents the basis of a theory of entanglement. We begin with a classical theory of entangled discrete measures in Section~1. Section~2 treats quantum mechanics and discusses the statistics of bounded operators on a Hilbert space in terms of context coefficients. In Section~3 we combine the work of the first two sections to develop a general theory of entanglement for quantum states. A measure of entanglement called the entanglement number is introduced. Although this number is related to entanglement robustness, its motivation is not the same and there are some differences. The present article only involves bipartite systems and we leave the study of multipartite systems for later work., Comment: 20 pages

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.

Plant mitochondrial genomes vary widely in size. Although many plant mitochondrial genomes have been sequenced and assembled, the vast majority are of angiosperms, and few are of gymnosperms. Most plant mitochondrial genomes are smaller than a megabase, with a few notable exceptions. We have sequenced and assembled the complete 5.5-Mb mitochondrial genome of Sitka spruce (Picea sitchensis), to date, one of the largest mitochondrial genomes of a gymnosperm. We sequenced the whole genome using Oxford Nanopore MinION, and then identified contigs of mitochondrial origin assembled from these long reads based on sequence homology to the white spruce mitochondrial genome. The assembly graph shows a multipartite genome structure, composed of one smaller 168-kb circular segment of DNA, and a larger 5.4-Mb single component with a branching structure. The assembly graph gives insight into a putative complex physical genome structure, and its branching points may represent active sites of recombination.

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.

Self-testing refers to a device-independent way to uniquely identify an unknown quantum device based only on the observed statistics. Earlier results on self-testing of multipartite state were restricted either to Dicke states or Graph states. In this paper, we propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of $W$ state and $GHZ$ state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical bases, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semidefinite programming (SDP) in high precision.

Complex systems in the real world inevitably suffer from unpredictable perturbations, which can trigger system disasters, wreaking significant economical losses. To exploit the robustness of complex systems in the face of disturbances is of great significance. One of the most useful methods for system robustness analysis comes from the field of complex networks characterized by percolation theories. Many percolation theories, therefore, have been developed by researchers to investigate the robustness of diverse complex networks. Nevertheless, extant percolation theories are primarily devised for multilayer or interdependent networks. Little endeavor is dedicated to systems with multipartitivity structures, that is, multipartite networks, which are an indispensable part of complex networks. This paper fills this research gap by theoretically examining the robustness of multipartite networks under random or target node attacks. The generic percolation theory for robustness analysis of multipartite networks is accordingly put forward. To validate the correctness of the proposed percolation theory, we carry out simulations on computer-generated multipartite networks with Poisson degree distributions. The results yielded by the proposed theory coincide well with the simulations. Both theoretical and simulation results suggest that complex systems with multipartitivity structures could be more robust than those with multilayer structures. Accepted version

The article examines young people’s attitudes towards enterprise, comparing prosperous and deprived neighbourhoods and two UK cities. Corpus linguistics analysis identified multi-layered attitudes and variations in how place prosperity and city affect attitudes. High interest in enterprise was associated with weaker place attachment and reduced social embeddedness. Young adults from prosperous neighbourhoods delegitimised other’s enterprises; the ‘deprived’ sub-corpus included more fluid notions of enterprise legitimacy. Liverpool accounts contained stronger discursive threads around self-determination; Bradford accounts included greater problematising of entrepreneurship versus employment. An original Multipartite Model of Attitudes to Enterprise is presented consisting of four layers: attitudes to enterprise generally, attitudes legitimising particular forms of enterprise, attitudes to enterprise related to place and attitudes to enterprise related to self. The conclusion explains why policies and research need to be fine-grained and avoid uni-dimensional conceptualisations of attitudes to enterprise, or deterministic arguments relating entrepreneurship to specific types of places or backgrounds.

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

We propose a scheme for a switchable coupling between several Cooper-pair boxes in the charge regime. The switch is embodied in a SQUID element contained in the center conductor of a transmission-line resonator. Altering the flux bias through the SQUID allows for changing the effective resonator length. Thus the position of the nodes and anti-nodes of the relevant eigenmodes changes and leads to a variable interaction strength between qubit and cavity vacuum field. For the coupled qubits the interaction is dispersive. An example for the application of this switch is the generation of multipartite entangled states for three and four charge qubits. Although used as a discrete switch in the present proposal, the combined system of SQUID module integrated into the transmission line may be operated continuously as well.

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

Multipartite viruses have a segmented genome, with each segment encapsidated separately. In all multipartite virus species for which the question has been addressed, the distinct segments reproducibly accumulate at a specific and host-dependent relative frequency, defined as the ‘genome formula’. Here, we test the hypothesis that the multipartite genome organization facilitates the regulation of gene expression via changes of the genome formula and thus via gene copy number variations. In a first experiment, the faba bean necrotic stunt virus (FBNSV), whose genome is composed of eight DNA segments each encoding a single gene, was inoculated into faba bean or alfalfa host plants, and the relative concentrations of the DNA segments and their corresponding messenger RNAs (mRNAs) were monitored. In each of the two host species, our analysis consistently showed that the genome formula variations modulate gene expression, the concentration of each genome segment linearly and positively correlating to that of its cognate mRNA but not of the others. In a second experiment, twenty parallel FBNSV lines were transferred from faba bean to alfalfa plants. Upon host switching, the transcription rate of some genome segments changes, but the genome formula is modified in a way that compensates for these changes and maintains a similar ratio between the various viral mRNAs. Interestingly, a deep-sequencing analysis of these twenty FBNSV lineages demonstrated that the host-related genome formula shift operates independently of DNA-segment sequence mutation. Together, our results indicate that nanoviruses are plastic genetic systems, able to transiently adjust gene expression at the population level in changing environments, by modulating the copy number but not the sequence of each of their genes.