Matrix description of the complete topology of three-dimensional cells

A new, efficient method based on a series of matrices is introduced to completely describe the detailed topology of individual domains and their topology evolution in three-dimensional cellular structures. With this approach, we found a lot of new topological grain forms which are never reported before, i.e., there are total 8 and 32 topological forms for 7- and 8-faced grains respectively, other than the reported 7 and 27. This method is proved to be a practical tool to predict all possible grain forms efficiently. Moreover, a connectivity index of grain forms serves as a new convenient differentiator of different multicellular structures.

grain class, 32 forms in 8-faced grain class and 131 forms in 9-faced grain class respectively (See details in the Supplemental Material), which proved that a lot of grain forms are left out in the previous findings 13 .

Methods
We consider n to be the faces number of a grain, p the vertices number, and q the edges number. Euler's Law relates the faces, edges, and vertices of a polyhedron as (n − q + p = 2) 17 . Under the restrictions of surface tension in three-dimensional cellular structures, three boundary interfaces meet at an edge and four edges meet at a vertex. Thus, for an individual grain, each edge has two vertices and each vertex radiates three edges, we can get (2q = 3p) 3 . We define the grain topological face adjacency matrix as the matrix A f(e) = [α ij ] n × n , in which α ij = k if the face i and j have k joint edges, and α ij = 0 if i and j have no joint edge or i = j. Figure 1 shows the Schlegel diagram of a 5-faced grain and its face adjacency matrix.
Topological properties like the number of grain faces, the number of edges per face, and the neighboring relations among different faces are involved in this matrix and can be calculated efficiently from matrix operation. The face adjacency matrix contains all the information of the correlations among grain faces, which can be used to completely describe the grain topology. We can also define the vertex adjacency matrix and edge adjacency matrix, however, they are not the complete topology description for grains whose Schlegel diagrams are not 3-connected graph.
The grains with same topological forms have isomorphic Schlegel diagram and permutation-similar A f(e) . The spectra (i.e. the set of eigenvalues together with their multiplicities) of their A f(e) are identical. In spectral graph theory, the (adjacency) spectrum of a graph, which is the spectrum of its adjacency matrix, is one of its topological invariants 16 . Thus, the spectrum of A f(e) can be used to describe the grain topology.
The Laplacian matrix of a graph is defined as the degree matrix minus the adjacency matrix. For grain forms, is a diagonal matrix in which the entries on main diagonal equal to the edge number of the grain faces. The second-smallest eigenvalue of the Laplacian matrix, λ 2 , is a very important parameter to describe the graph characteristics, which is a topological index and called algebraic connectivity 16 .

Results and Discussions
Topology characteristics of grain forms. Lazar 10 presented eight most common grain topologies in grain growth and Poisson-Voronoi microstructure and observed that grain growth microstructure favor certain highly symmetric grain topologies relative to the Poisson-Voronoi microstructure. Table 1. listed their corresponding λ 2 values. It can be seen that the most frequent grain forms in grain growth microstructure always has large λ 2 values among all possible forms in a given face class. For 7-, 8-, 9-and 10-faced grains in grain growth microstructures, the most frequent grain topologies corresponds to 1 st , 2 nd ,1 st and 1 st largest λ 2 values in their total 8, 32, 131 and 723 forms, respectively. This reveals that some properties related to large λ 2   10 . Their p vectors, number of faces, algebraic connectivity λ 2 , and the ranking of λ 2 in total forms of each face class are included.
are favored by grain growth, termed as algebraic connectivity preferred tendency (large λ 2 preferred), besides the highly symmetric form preferred tendency observed in Lazar's study 10 .
In spectral graph theory, large value of λ 2 represents good connectivity, expansion and randomness properties of a graph 16 . Apparently, λ 2 is influenced by not only the topological symmetry but also the edge distribution of grain faces, etc. For highly symmetric topological forms, if they have a narrow edge distribution of faces, i.e., the number of edges of each face approximates the average edges per face, the values of λ 2 are large.
Actually, a narrow edge distribution of faces for a topological form is favorable in grain growth. Grain growth involves the form transition from higher face class to lower one by the disappearing of triangular faces. If a grain has more triangular faces, it increased the probability of face loss and transfer down the topological ladder to a disappearing tetrahedron to maintain grain growth 13 . And if a grain has no triangular face, it should employ edge-switch transitions to create triangular faces before decreasing to lower face classes. For a larger-edged face, the process of transferring down its edges ladder to ultimately a triangular face is complex relative to the smaller-edged faces. From the view of edge evolution, a larger-edged face of a grain is more stable than a smaller-edged faces. On the other hand, if one of grain faces increases its stability by increasing its edges, the edges of other faces are sure to decrease under the constraint of Euler's equation. Therefore, the realistic way to increase the stability of a topological form is to modulate the edges of each face to approximate the average edges per grain face, which result in a narrow edge distribution of faces and a large λ 2 value.
To make more clear of the difference between the algebraic connectivity and symmetry of topological forms, Fig. 2 lists all the topological forms of 7-faced grains together with the λ 2 values and the order S of the symmetry forms. The λ 2 value of each form decreased from left to right. The grain with p vector (0005200) has five 4-edged faces and two 5-edged faces, approximating to the average edges per face 4.2 for 7-faced grain class, has the largest λ 2 value 3.382. The right-hand three forms have the smallest λ 2 values, however, they have good symmetry, especially the most right hand one. These three forms are unstable and likely occur only as a transient state in the grain growth process, although they have high symmetry.
With the increase of triangular faces, the deviation degree from the average edges per face will increase, which resulted in low algebraic connectivity for grains. From the algebraic connectivity preferred tendency, these forms will have a less frequency in grain growth microstructures. This is in accord with Patterson's result 13 that the highest frequency forms have few or no triangular faces, while those with the most triangular faces are present the least.
The difference between the grain growth and Poisson-Voronoi microstructure can also be observed with respect to λ 2 value in Table 1. The λ 2 values of the most frequent grain topologies in grain growth microstructures are substantially larger than the corresponding ones in Poisson-Voronoi microstructures. This reveals that the availability of the matrix topology description can be used as another tool for distinguishing between fundamental characteristics of different cellular microstructures, besides the reported order of symmetry 10 of grain topology in Weinberg-vector method.
Grain form evolution in grain growth. An important application of the matrix topology description is the highly efficient analysis of the grain topological form evolution during the grain growth process. The fundamental topological events 13 occurring in three-dimensional grain growth include: Type I -loss of a grain face through disappearance of a neighboring tetrahedron, Type II-grain encounter between grains not formerly touching and Type III -separation of grains sharing a face.
Steele 18 used Schlegel diagrams and the topological events to study the collapsing polycrystal grains, illustrating the possible topological forms of grains by means of downward and horizontal transitions from 8 to 4 faces. Patterson 13 expands Steele's original analysis to include other possible grain forms resulting from uphill  and horizontal transitions. However, it is difficult to work out all possible grain forms only by hands with the help of Schlegel diagrams and the topological events. That is part of the reasons why the above studies on topological forms are incomplete and terminated at 8-faced grains.
As the grain form evolves based on the topology events, the grain topological matrices will transform at the same time. We can use matrix operation to analyze the possible grain forms in grain growth efficiently. Through the fundamental topological events, a grain topological form would experience 3 possible changes: (1) losing a triangular face, (2) gaining a triangular face and (3) rearrangement of faces 13 . This can be reflected onto grain's Schlegel diagram (See Fig. 3). Now we give the matrix description of the change (1) losing a triangular face as an example.
For convenience, three incidence matrices are constructed depending on the vertices, edges and faces of grain forms.
The generalized elementary matrices are used in the matrix operation, which include row-deleting matrix (E i , deleting row i of identity matrix, I) and row-inserting matrix (U i , inserting a all 0's row between row (i-1) and row For the change of (1) losing a triangular face, we consider that the mth face of an n-faced grain is a triangular face. Losing this face corresponds to the operation of deleting row m and the three columns with value "1" in M n f−e . It is convenient to use the generalized elementary matrices to do matrix operation. That is The most realistic means for determining possible grain forms is through the processes causing face-gain or loss in normal grain growth, i.e. encounter, separation and horizontal edge switch transitions within the same face class 13 . We begin with the lone tetrahedron in face class 4 and determine the possible higher forms gradually by uphill and horizontal topology transformations. Firstly, we place a triangular face at any one of the vertices of a grain form in lower face class to get a new grain form in higher face class. Secondly, we do the edge-switch transitions for this form and some other new forms would be created. Thirdly, we do edge-switch transitions again for all forms created in the previous step to find more new forms. The third step was repeated until there are no new forms created. Figure 4 shows the flow chart of the process of the generation of all the possible grain forms. The spectrum of A f(e) is used to distinguish whether a created form is a new one. Although having the same spectrum is only the necessary condition for two graphs to be isomorphic (i.e., the two forms are the same), it is a practice criterion because there have been found very few examples of non-isomorphic cubic graphs that have the same spectrum 19 .
This method is able to generate all the possible grain forms, in that any one of f-faced forms can be transformed to all the other forms with f faces by the horizontal edge switch transitions, i.e., rearrangement of faces. The detailed proof is presented in the Supplemental Material and the path of the transition of any two 7-faced  grain forms is also included as an example. Figure 5 shows the sketch of the generation of 5-and 6-faced forms from the beginning 4-faced forms.
The matrix description of grain form evolution in grain growth was implemented by developing a program written in Scilab 5.5.1 (free and open source software for numerical computation, Scilab Enterprises, available from: http://www.scilab.org). Table 2 lists the number of topological forms calculated by this program with a 4 × 3.40 GHz CPU, 16 G RAM computer in an acceptable period of time (e.g., 5 hours for 13-faced grains). For comparison, some similar results are also included. The enumeration of 3-regular simple (3-connected) and band-faced (2-connected but not 3-connected) planar graphs using a Monte Carlo algorithm is done by Keller 15 . The number of our calculated grain forms with 3-connected Schlegel diagrams in this work is the same as Keller's for grains with 13 and fewer faces, but more larger than Keller's results for grains with 14 faces. Especially, the number of our calculated band-faced planar graphs is larger than Keller's results for grains with 8 and larger faces. This result indicates that many grain forms are not found by Monte Carlo method of which the reason is still clear. The generated simple graphs by plantri software are a little more than our and Keller's results for grains with 13 and 14 faces, which is possibly because of the error in judging two same grain forms in different methods and the exact reason still awaits further analysis. The plantri software cannot generate band-faced planar graphs which limits its use in the grain growth studies.
We plot all the Schlegel diagrams for 8-and 9-faced grains in the Supplementary Material. Figure 6 lists 5 new forms found in this work beyond the known 27 forms for 8-faced grains as reported 13 . They all belong to band-faced topological grain forms and are observed to have characteristic pairs of contiguous three-edged faces. These three-edged face pairs are supposed to be unstable and decompose rapidly 13 in grain growth process, resulting in relative low frequency of these grain forms in microstructures. It can also be seen that these forms in Fig. 5 have very high symmetry but wide edge distribution of grain faces. This further indicates that high symmetry is possibly a necessary condition for high frequency of occurrence of a certain grain form but not a sufficient condition. Those forms that occur more frequently than others normally have the characteristics of high symmetry plus narrow edge distribution of grain faces, which can be described by the new topology index found in this work-algebraic connectivity.

Conclusion
In conclusion, we have introduced an efficient method to completely describe grain topologies. The application of this method to the grain form analysis has shown that the grain growth microstructure favors topological forms with large algebraic connectivity relative to the Poisson-Voronoi microstructure. This suggests that not only highly symmetric topological forms, but also forms with narrow edge distribution of faces are preferred in grain growth microstructure. The algebraic connectivity of grain forms has been proved to be a new convenient differentiator of different multicellular structures. Further, the matrix description of grain topology can serves as a new practical tool to predict the possible grain forms in three-dimensional grain growth efficiently.  Table 2. Summary of the grain forms generated using our matrix description method, Monte Carlo method 15 and plantri. 15