Common Edges Research Articles

Computing Hive Plots: A Combinatorial Framework

Hive plots are a graph visualization style placing vertices on a set of radial axes emanating from a common center and drawing edges as smooth curves connecting their respective endpoints. In previous work on hive plots, assignment to an axis and vertex positions on each axis were determined based on selected vertex attributes and the order of axes was prespecified. Here, we present a new framework focusing on combinatorial aspects of these drawings to extend the original hive plot idea and optimize visual properties such as the total edge length and the number of edge crossings in the resulting hive plots. Our framework comprises three steps: (1) partition the vertices into multiple groups, each corresponding to an axis of the hive plot; (2) optimize the cyclic axis order to bring more strongly connected groups near each other; (3) optimize the vertex ordering on each axis to minimize edge crossings. Each of the three steps is related to a well-studied, but NP-complete computational problem. We combine and adapt suitable exact and heuristic algorithmic approaches, implement them as an instantiation of our framework, and show in a case study how it can be applied in a practical setting. Furthermore, we conduct computational experiments to gain further insights regarding the algorithmic choices of our framework. The code of the implementation and a prototype web application can be found on OSF.

Multinuclear solid state NMR spectroscopy of ternary rare-earth silicides RET 2Si2 and germanides LaT 2Ge2 (RE = Sc, Y, La, Lu; T = Fe, Co, Ni, Cu, Ru, Rh, Pd, Ag, Os, Ir, Pt, Au)

Abstract A series of ternary rare earth – transition metal – tetrelides RET 2 Tt 2 (RE = Sc, Y, La, Lu; T = Fe, Co, Ni, Cu, Ru, Rh, Pd, Ag, Os, Ir, Pt, Au; Tt = Si, Ge) was synthesized by arc melting of the elements and subsequent annealing. The samples were characterized by powder X-ray diffraction and in addition, the structures of REOs2Si2 (RE = Y, La, Lu), LaAu2Si2, LaAg2Ge2 and LaAu2Ge2 were refined from single crystal X-ray diffractometer data. The tetrelides crystallize with the ThCr2Si2 type (I4/mmm) except the platinum compounds which adopt the klassengleiche superstructure of the CaBe2Ge2 type (P4/nmm). The transition metal atoms have tetrahedral tetrel coordination and the tetrahedra condense to layers via common edges. The stacking of these layers leads to Tt−Tt bonds in the ThCr2Si2 type phases and heteroatomic T−Tt bonds in the CaBe2Ge2 type phases. The rare earth atoms fill larger cages within these three-dimensional networks (coordination number 16 with RE@T 8 Tt 8) with site symmetries 4/mmm (ThCr2Si2 type) and 4mm (CaBe2Ge2 type). Systematic multinuclear solid state NMR spectroscopic investigations allowed observing the effect of the involved rare-earth metal, transition metal and tetrel group element, respectively. In particular, 29Si isotropic resonance shifts can be predicted from element-specific increments and interatomic Si–Si bonding interactions manifest themselves in axially symmetric magnetic shielding anisotropies.

Community detection by influential nodes based on random walk distance

SummaryCommunity detection is a key feature that can be used to extract useful information from networks. In most studies, the same static algorithms are used on real‐time snapshots of the dynamic network. Such an action increases the calculations and the time taken for the clustering operation. The idea of community detection is based on the fact that communities are formed around a node that is more popular and influential. Therefore, in the proposed algorithm, first, several snapshots are received from the network, then for the first snapshot, the influence of each node is calculated using the influence function based on the random walk distance between nodes. Then by selecting k nodes with higher influence, network communities are formed and other nodes belong to the community with the most common edges. In the next step, the next snapshots will be received and then the communities will be updated. Then K nodes with higher influence are selected and their community is created if needed. The proposed method has been compared with state‐of‐the‐art algorithms, and the results show that proposed method has been able to have a suitable performance in the uniformity of communities and also the speed of community formation.

A polynomial‐time approximation scheme for the maximal overlap of two independent Erdős–Rényi graphs

AbstractFor two independent Erdős–Rényi graphs , we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial‐time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by‐product, we prove that the maximal overlap is asymptotically for with some constant .

A novel method for PET connectomics guided by fibre-tracking MRI: Application to Alzheimer's disease.

This study introduces a novel brain connectome matrix, track-weighted PET connectivity(twPC) matrix, which combines positron emission tomography (PET) and diffusion magnetic resonance imaging data to compute a PET-weighted connectome at the individual subject level. The new method is applied to characterise connectivity changes in the Alzheimer's disease (AD) continuum. The proposed twPC samples PET tracer uptake guided by the underlying white matter fibre-tracking streamline point-to-point connectivity calculated from diffusion MRI (dMRI). Using tau-PET, dMRI and T1-weighted MRI from the Alzheimer's Disease Neuroimaging Initiative database, structural connectivity (SC) and twPC matrices were computed and analysed using the network-based statistic (NBS) technique to examine topological alterations in early mild cognitive impairment (MCI), late MCI and AD participants. Correlation analysis was also performed to explore the coupling between SC and twPC. The NBS analysis revealed progressive topological alterations in both SC and twPC as cognitive decline progressed along the continuum. Compared to healthy controls, networks with decreased SC were identified in late MCI and AD, and networks with increased twPC were identified in early MCI, late MCI and AD. The altered network topologies were mostly different between twPC and SC, although with several common edges largely involving the bilateral hippocampus, fusiform gyrus and entorhinal cortex. Negative correlations were observed between twPC and SC across all subject groups, although displaying an overall reduction in the strength of anti-correlation with disease progression. twPC provides a new means for analysing subject-specific PET and MRI-derived information within a hybrid connectome using established network analysis methods, providing valuable insights into the relationship between structural connections and molecular distributions. PRACTITIONER POINTS: New method is proposed to compute patient-specific PET connectome guided by MRI fibre-tracking. Track-weighted PET connectivity (twPC) matrix allows to leverage PET and structural connectivity information. twPC was applied to dementia, to characterise the PET nework abnormalities in Alzheimer's disease and mild cognitive impairment.

Ca3Ru3Zn13 – a zinc-rich intermetallic phase composed of closest-packed layers

Abstract Ca3Ru3Zn13 was synthesized from the elements in a tantalum tube in a muffle furnace. The sample was annealed to 1123 K followed by slow cooling to enhance crystal growth. Ca3Ru3Zn13 was characterized through a Guinier powder pattern. The structure was refined from single crystal X-ray diffractometer data: new type, Pbcn, a = 522.92(5), b = 1792.72(14), c = 2789.6(2) pm, wR2 = 0.0590, 3451 F 2 values, 177 variables. One can describe the Ca3Ru3Zn13 structure by a stacking of closest packed layers of compositions Ca6Zn10, Ca6Zn12 and Ru6Zn15. The calcium atoms have high coordination numbers: Ca1@Zn13Ru3Ca2, Ca2@Zn15Ru4Ca, Ca3@Zn15Ru4Ca and Ca4@Zn15Ru4Ca. The shortest interatomic distances occur between the ruthenium and zinc atoms (255–273 pm). The Ru1@Zn8, Ru2@Zn10 and Ru3@Zn8 polyhedra condense via common edges to the [Ru3Zn13] substructure which hosts the calcium atoms. The zinc substructure consists of 15 crystallographically independent zinc sites with a broader range of Zn–Zn distances (255–307 pm).

The germanides ScTGe2 (T = Fe, Co, Ru, Rh) – crystal chemistry, 45Sc solid-state NMR and 57Fe Mössbauer spectroscopy

Abstract The TiMnSi2-type (space group Pbam) germanides ScTGe2 (T = Fe, Co, Ru, Rh) were synthesized from the elements by arc-melting. Single crystals were grown by annealing sequences of the arc-melted buttons in an induction furnace. The structures of ScFeGe2, ScRuGe2 and ScRhGe2 were refined from single-crystal X-ray diffraction data. In ScRuGe2, the ruthenium atoms have distorted octahedral germanium coordination (242–268 pm Ru–Ge). Three trans-face-sharing octahedra form a sub-unit which is condensed via common edges in c direction and connected via common corners with four adjacent blocks, forming a three-dimensional [RuGe2 type] substructure. The two crystallographically independent scandium sites have coordination numbers 15 (Sc1@Ge8Ru4Sc3) and 17 (Sc2@Ge7Ru6Sc4). Electronic band structure calculations for ScCoGe2 and ScRuGe2 show a net charge transfer from the scandium to the transition metal and germanium atoms, leading to a description with polyanionic networks Sc δ+[TGe2]δ−. The two crystallographically independent Sc sites are easily distinguishable by 45Sc magic-angle spinning (MAS)-NMR spectroscopy. Isotropic chemical shift values and nuclear electric quadrupolar interaction parameters were deduced from an analysis of the triple-quantum (TQ)-MAS NMR spectra. The electric field gradient parameters deduced from these experiments are in good agreement with quantum-chemical calculations using the Wien2k code. Likewise, the two crystallographically independent iron sites in ScFeGe2 could be discriminated in the 57Fe Mößbauer spectra through their isomer shifts and quadrupole splitting parameters: δ = 0.369(1) mm s−1 and ∆E Q = 0.232(2) mm s−1 for Fe1 and δ = 0.375(2) mm s−1 and ∆E Q = 0.435(4) mm s−1 for Fe2 (data at T = 78 K).

Effect of Twin Circular Underground Voids on Bearing Capacity of Strip Foundations

Underground voids can significantly alter the stress and strain conditions of the surrounding soil mass, consequently affecting the stability of buildings located on the surface. Numerous numerical and experimental methods have been employed to investigate the effects of voids on nearby structures. In this research, a mixed lower bound-finite element method is utilized to study the stability of strip foundations situated above twin continuous voids. The soil behavior is modeled using the Mohr-Coulomb yield function along with an associated flow rule. Three-noded triangular stress elements are used to mesh the stress field, considering stress discontinuities at the common edges of adjacent elements. The study assumes that the two voids are identical in terms of radius, distance from the footing centerline, and depth from the ground level. The primary geometrical parameters considered in the analysis are the location and radius of the voids, as well as the strength parameters of the soil. By investigating the changes in bearing capacity concerning these factors, practical charts are proposed to aid in understanding the stability of the foundation. The results of the current study reveal the existence of an unsafe zone beneath the foundations, where the impact of the twin voids on bearing capacity reduction can be significant. Understanding these effects is crucial for design and construction of structures in areas prone to underground voids and also to decide where to bore infrastructure tunnels in order to have minimum effect on stability of the above buildings.

Crystal Chemistry and Structural Complexity of the Uranyl Molybdate Minerals and Synthetic Compounds

This paper reviews not the largest, but at the same time quite an interesting, group of natural and synthetic uranyl molybdate compounds. Nowadays, nine minerals of U and Mo are known, but the crystal structures have only been reported for five of them. Almost an order of magnitude more (69) synthetic compounds are known. A significant discrepancy in the topological types for natural and synthetic phases is shown, which is most likely due to elevated temperatures of laboratory experiments (up to 1000 °C), while natural phases apparently grow at significantly lower temperatures. At the same time, the prevalence of dense topologies (with edge-sharing interpolyhedral linkage) among natural phases can be noted, which is fully consistent with other recently considered mineral groups. Uranyl molybdates demonstrate several similarities with compounds of other U-bearing groups; however, even topological matches do not lead to the appearance of completely isotypic compounds. Structural complexity calculations confirm, in general, crystal chemical observations. Considering the prevalence of dense structures in which coordination polyhedra of uranium and molybdenum are connected through common edges as well as framework architectures, one can expect a less significant influence of interlayer species on the formation of the crystal structure than the main U-bearing complexes. The more structural complexity of the uranyl molybdate units, the more complex of the entire crystal structure is. In addition, there is a tendency for complexity to increase with increasing density of the complex; the simplest structures are vertex-shared, while the complexity increases with the appearance of common edges.

Flexible Kokotsakis Meshes with Skew Faces: Generalization of the Orthodiagonal Involutive Type

In this paper, we introduce and study a remarkable class of mechanisms formed by a 3 × 3 arrangement of rigid quadrilateral faces with revolute joints at the common edges. In contrast to the well-studied Kokotsakis meshes with a quadrangular base, we do not assume the planarity of the quadrilateral faces. Our mechanisms are a generalization of Izmestiev’s orthodiagonal involutive type of Kokotsakis meshes formed by planar quadrilateral faces. The importance of this Izmestiev class is undisputed as it represents the first known flexible discrete surface – T-nets – which has been constructed by Graf and Sauer. Our algebraic approach yields a complete characterization of all flexible 3 × 3 quad meshes of the orthodiagonal involutive type up to some degenerated cases. It is shown that one has a maximum of 8 degrees of freedom to construct such mechanisms. This is illustrated by several examples, including cases which could not be realized using planar faces. We demonstrate the practical realization of the proposed mechanisms by building a physical prototype using stainless steel. In contrast to plastic prototype fabrication, we avoid large tolerances and inherent flexibility.

The HfFe2Si2 type silicides ScT 2Si2 (T = Ru, Rh, Os) – structure and solid-state 29Si/45Sc NMR spectroscopy

Abstract The silicides ScT 2Si2 (T = Ru, Rh, Os) were synthesized by arc-melting of the elements and subsequent annealing in sealed silica ampoules. They crystallize with the rarely observed HfFe2Si2 type structure, space group Pbcm. The structures of ScRu2Si2 (a = 761.64(4), b = 730.70(6), c = 521.07(6) pm, wR = 0.0314, 633 F 2 values, 31 variables) and ScOs2Si2 (a = 771.10(8), b = 736.68(7), c = 521.88(5) pm, wR = 0.0479, 623 F 2 values, 31 variables) were refined from single crystal X-ray diffractometer data. The refinements showed small degrees of Ru/Si respectively Os/Si mixing on one 4c site, leading to the refined compositions ScRu1.96(1)Si2.04(1) and ScOs1.91(1)Si2.09(1). The monomeric building units in both structures are two slightly distorted, crystallographically independent RuSi5 respectively OsSi5 square pyramids, which are condensed via common edges. The resulting densely packed [Ru2Si2] and [Os2Si2] networks leave voids for the scandium atoms with coordination number 18: Sc@Si8Ru8Sc2 and Sc@Si8Os8Sc2. Temperature dependent magnetic susceptibility measurements of ScRu2Si2 and ScOs2Si2 indicate Pauli paramagnetism. Solid-state 29Si and 45Sc static and MAS NMR spectroscopy reveal significant Knight shifts and a strong influence of the T/Si mixing on the experimental NMR line shapes. A larger overall 29Si magnetic shift for ScRu2Si2 compared to ScOs2Si2, although with a negative sign for one of the 29Si resonances, suggests stronger paramagnetic effects for ScRu2Si2, in agreement with the magnetic susceptibility measurements.

RbEr2AsS7: A Rubidium-Containing Erbium Sulfide Thioarsenate(III) with (S2)2− Ligands According to RbEr2S(S2)[AsS2(S2)

The new rubidium-containing erbium sulfide thioarsenate(III) with the structured formula RbEr2S(S2)[AsS2(S2)] was obtained from the syntheses of elemental erbium (Er), arsenic sesquisulfide (As2S3) and rubidium sesquisulfide (Rb2S3) with elemental sulfur (S) at 773 K as transparent, orange, needle-shaped crystals. RbEr2AsS7 crystallizes monoclinically in the space group C2/c with a = 2339.86(12) pm, b = 541.78(3) pm, c = 1686.71(9) pm and β = 93.109(3) ° for Z = 8. The crystal structure features complex [AsS2(S2)]3− anions with two S2− anions and a (S2)2− disulfide dumbbell coordinating end-on as ligands for each As3+ cation. Even outside the ligand sphere of As3+, S2− and (S2)2− can be found as sulfide anions. Two distinct Er3+ cations are surrounded by either nine or seven sulfur atoms. The [ErS9] polyhedra are corner- and face-connected, while the [ErS7] units share common edges, both building chains along [010]. These different chains undergo edge connectivity with each other, resulting in the formation of corrugated layers, which are held together by Rb+ in chains of condensed [RbS9] polyhedra. So, a three-dimensional network is generated, offering empty channels along [010] apt to take up the As3+ lone-pair cations. Wavelength-dispersive X-ray spectroscopy verified a molar Rb:Er:As:S ratio of approximately 1:2:1:7 and diffuse reflectance spectroscopy showed the typical f–f transitions of Er3+, while the optical band gap was found to be 2.42 eV.

Synthesis and Crystal Structure of the Europium(II) Hydride Oxide Iodide Eu5H2O2I4 Showing Blue-Green Luminescence.

As the first europium(II) hydride oxide iodide, dark red single crystals of Eu5H2O2I4 could be synthesized from oxygen-contaminated mixtures of EuH2 and EuI2. Its orthorhombic crystal structure (a = 1636.97(9) pm, b = 1369.54(8) pm, c = 604.36(4) pm, Z = 4) was determined via single-crystal X-ray diffraction in the space group Cmcm. Anion-centred tetrahedra [HEu4]7+ and [OEu4]6+ serve as central building blocks interconnected via common edges to infinite ribbons parallel to the c axis. These ribbons consist of four trans-edge connected (Eu2+)4 tetrahedra as repetition unit, two H--centred ones in the inner part, and two O2--centred ones representing the outer sides. They are positively charged, according to ∞1{[Eu5H2O2]4+}, to become interconnected and charge-balanced by iodide anions. Upon excitation with UV light, the compound shows blue-green luminescence with the shortest Eu2+ emission wavelength ever observed for a hydride derivative, peaking at 463 nm. The magnetic susceptibility of Eu5H2O2I4 follows the Curie-Weiss law down to 100 K, and exhibits a ferromagnetic ordering transition at about 10 K.

Structural and computational studies on a quenchable high-temperature polymorph of magnesium tungstate (MgWO4-II)

Single-crystals of a quenchable high-temperature polymorph of magnesium tungstate (MgWO4-II) have been grown using the flux method. Polycrystalline material of the same compound could be obtained from solid-state reactions performed at 1200 °C. Basic crystallographic data of the previously unknown modification are as follows: space group P1‾, a = 6.5525(6) Å, b = 7.5883(7) Å, c = 7.6976(6) Å, α = 119.064(9)°, β = 95.545(7)°, γ = 107.645(8)°, V = 304.84(5)3 Å and Z = 4. The crystal structure was solved from single-crystal diffraction using direct methods and refined to a residual value of R1 = 2.16%. Both cation species exhibit an octahedral oxygen coordination environment. By sharing common edges and corners, the octahedra form a three-dimensional network, which can be built from infinite rod-like elements running along [010] having a 2 × 2 octahedra wide cross section. MgWO4-II is topologically equivalent to the VO2(HT) structure-type. A detailed analysis of the relationships with other ABO4-compounds is presented. Solid-state characterization has been supplemented by micro-Raman spectroscopy. Interpretation of the spectroscopic data, including the allocation of the bands to certain vibrational species, has been aided by DFT-calculations. The thermal expansion tensor between ambient temperature and about 700 °C has been determined. The calculations indicate quasi-two-dimensional expansion behavior.

A New Mineral Hanauerite, AgHgSI, and Common Crystal Chemical Features of Natural Mercury Sulphohalides

A new mineral, hanauerite, ideally AgHgSI, was found in the oxidation zone of Ag- and Hg-bearing ores at two old, abandoned mines in Rhineland-Palatinate, Germany. In a holotype specimen originating from the Schöne Aussicht Mine, Dernbach, Westerwald, it is associated with plumbogummite–hinsdalite series of minerals and goethite. In cotype from the Friedrichssegen Mine, Bad Ems, it is associated with perroudite, goethite, and quartz. At both localities, hanauerite occurs as a prismatic crystal up to 0.15 mm long and up to 0.02 mm thick. The mineral is yellow, transparent, with an adamantine lustre. It is brittle, and cleavage was not observed. The calculated density values are 6.671 and 6.575 g cm−3 for holotype and cotype, respectively. The empirical formulae calculated (from electron microprobe data) based on the sum of all atoms = 4 apfu are Ag0.95Hg1.00S1.01(I0.83Br0.19Cl0.03)Σ1.05 for holotype and Ag0.97Hg0.97S1.05(I0.76Br0.25)Σ1.01 for cotype. Hanauerite is orthorhombic, space group Pmma; the unit cell parameters (from single-crystal X-ray diffraction data; holotype/cotype) are: a = 9.932(2)/9.9256(8), b = 4.6219(19)/4.6209(2), c = 9.891(4)/9.9006(4) Å, V = 454.0(3)/454.19(5) Å3, and Z = 4. The crystal structure was studied on single crystals extracted from both holotype and cotype specimens; R1 = 0.0416 (holotype) and =0.0544 (cotype). In hanauerite, Hg2+ cations centre strongly distorted octahedra with two short Hg–S bonds (Hg and S atoms build “crankshaft-type” chains) and four strongly elongated Hg–I bonds. The Hg-centred octahedra are connected via common edges and faces to form corrugated layers; Ag+ cations are located between these layers. Hanauerite is named in honour of the German mineral collector Dr. Alfred Hanauer (1912–1988). The common crystal chemical features of mercury sulphohalide minerals are discussed.

Higher-order modeling of a thin-walled beam with a welded multicell cross-section and its application to welding line optimization

Thin-walled multicell structures observed in vehicle frames are partially welded to lower the welding process cost while maintaining their overall structural stiffness comparable to those of fully-welded multicell structures. A fast and reliable beam-based model for analyzing such structures is required in the concept stage of the design process. In this investigation, an analysis method for a thin-walled beam of a multi-cell cross-section partially welded along its axial direction is newly established using a higher-order beam theory (HoBT). Our approach uses the Lagrange multipliers to impose the partial welding condition along the common edges of multiple thin-walled closed sections using the three-dimensional field derived from a higher-order beam theory. The developed method in this study can be an accurate and effective alternative without full finite element analysis employing shell and solid elements. To find the optimal welding locations, we introduced binary design variables parameterized through their corresponding welding constraints to identify the status of welding or non-welding along a candidate welding line. A genetic algorithm is adopted to solve optimization problems. After establishing the analysis method and optimization technique, their validity was checked using double-cell and triple-cell sectioned beams and beam frame structures, including a bus structure. The stiffnesses of the structures with optimized partial welding lines (20%) drop less than 6% from those of fully welded counterparts.

Drawing Simultaneously Embedded Graphs with Few Bends

We study the problem of drawing simultaneously embedded graphs with few bends. We show that for any simultaneous embedding with fixed edges (SEFE) of two graphs, there exists a corresponding drawing realizing this embedding such that common edges are drawn as straight-line segments and each exclusive edge has a constant number of bends. If the common graph is biconnected and induced, a straight-line drawing exists. This yields the first efficient testing algorithm for simultaneous geometric embedding (SGE) for a non-trivial class of graphs.

Effect of CaO partial substituted by BaO on structure of aluminotitanate melts by molecular dynamics simulation

The effect of CaO partial substituted by BaO on the microstructure of aluminotitanate melts by molecular dynamics (MD) and spectroscopy, especially the alteration of Al2O3 and TiO2 amphoteric behavior. The MD simulation results indicated that as the BaO content increased from 15 to 18 wt%, TiO2 transformed into [TiO6] octahedral and Al2O3 transformed to alkaline. The BaO content increased above 18 wt%, and the continuous insufficient supply of cations further promoted the transition of TiO2 to low coordination reducing the potential connection between the common edges. The dissociation of Ba2+ from BaO supported the transformation of small molecular units into macromolecular units and improved the degree of polymerization of the melt. Raman spectra showed that when BaO concentration grew, the complexity of the [AlOx] polyhedra increased, promoting the transformation of TiO2 to alkaline [TiO6] first and then to acidic [TiO5], which is consistent with the results of the MD.

Crystal structure of barium dinickel(II) iron(III) tris-[orthophosphate(V)], BaNi2Fe(PO4)3.

The orthophosphate BaNi2Fe(PO4)3 has been synthesized by a solid-state reaction route and characterized by single-crystal X-ray diffraction and energy-dispersive X-ray spectroscopy. The crystal structure comprises (100) sheets made up of [Ni2O10] dimers that are linked to two PO4 tetra-hedra via common edges and vertices and of linear infinite [010] chains of corner-sharing [FeO6] octa-hedra and [PO4] tetra-hedra. The linkage of the sheets and chains into a framework is accomplished through common vertices of PO4 tetra-hedra and [FeO6] octa-hedra. The framework is perforated by channels in which positionally disordered Ba2+ cations are located.

The Polynomial Algorithm of Finding the Shortest Path in a Divisible Multiple Graph

In this paper, we study undirected multiple graphs of any natural multiplicity к > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of к linked edges, which connect 2 or (к + 1) vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of к linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. Divisible multiple graphs are characterized by a possibility to divide the graph into к parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. As for an ordinary graph, we can define the integer function of the length of an edge for a multiple graph and set the problem of the shortest path joining two vertices. Any multiple path is a union of к ordinary paths, which are adjusted on the linked edges of all multiple and multi-edges. In the article, we show that the problem of the shortest path is polynomial for a divisible multiple graph. The corresponding polynomial algorithm is formulated. Also we suggest the modification of the algorithm for the case of an arbitrary multiple graph. This modification has an exponential complexity in the parameter к.