Topological Coding and Topological Matrices Toward Network Overall Security

Bing Yao; Fei Ma; Hongyu Wang; Hui Sun; Jing Su; Meimei Zhao; Mingjun Zhang; Sihua Yang; Xiaohui Zhang; Xiaomin Wang

REVIEW 4 major objections 3 minor 27 references

Topological Coding and Topological Matrices Toward Network Overall Security

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single $3\times q$ Topcode-matrix can encode several distinct labelled graphs at once, letting one public matrix authenticate multiple private graph-based passwords.

desk verdict A matrix encoding of graph labelings with a catalog of examples, but the security claim is undefined and the main graphicability theorem is vacuous; desk-reject. read the letter →

arxiv 1909.01587 v2 pith:ZKYYSFAO submitted 2019-09-04 cs.IT math.IT

Bing Yao , Meimei Zhao , Xiaohui Zhang , Yarong Mu , Yirong Sun , Mingjun Zhang , Sihua Yang , Fei Ma

show 4 more authors

Jing Su Xiaomin Wang Hongyu Wang Hui Sun

This is my paper · ORCID

classification cs.ITmath.IT MSC 05C7805C5094A60

keywords Topcode-matrixTopsnut-gpwgraphicalpasswordtext-basedgraphlabellingmatrixgroupnetworksecurityHanzi-matrix

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Topological graphic passwords are usually pictures, but this paper recasts them as three-row matrices called Topcode-matrices: each column records one edge of a labelled graph as two vertex labels and one edge label. The paper argues that because many non-isomorphic labelled graphs share the same Topcode-matrix, one matrix can be a public key while several distinct graph passwords act as private keys. It then defines operations, including column exchange, row exchange, union addition, and additive or subtractive group laws, that turn collections of these matrices into algebraic groups, and it shows how the same matrices generate ordinary text-based passwords. If the security premise holds, the approach gives a way to protect networks with passwords that are simultaneously graphical, textual, and algebraic.

What carries the argument

The load-bearing object is the Topcode-matrix, a $3\times q$ array whose $i$-th column $(x_i, e_i, y_i)^T$ records one edge of a labelled graph: $x_i$ and $y_i$ are labels at the two ends and $e_i$ is the edge label derived from them by an evaluation rule. Its power is that many graphs can share the same matrix, because a matrix records labels but not the graph's topological arrangement; the same columns can be reassembled into different non-isomorphic Topsnut-gpws. The paper's operations run on this object: column-exchanging and $XY$-exchanging produce new matrices from the same graph, union-addition $\biguplus$ fuses several matrices into one, and additive or subtractive v-operations make every-zero groups in which any chosen matrix acts as the zero element. These operations carry the security argument, because the public object is a union matrix while the private objects are the constituent matrices and graphs.

What would settle it

Take a public union matrix built from two or more known Topcode-matrices and run a recovery attack: if the constituent columns can be separated by matching degree multiplicities, by solving $e_i=|x_i-y_i|$, or by exploiting the edge-label set, then the claimed one-vs-more security is refuted for that construction.

Watch

Extended reading notes

Core claim

The paper's central claim is that a Topcode-matrix can stand for many different labelled graphs at once, and that this one-to-many property is a security feature rather than an ambiguity. A Topcode-matrix is evaluated when an edge label $e_i$ is determined by its two end labels $x_i,y_i$ through a rule such as $e_i=|x_i-y_i|$ or a modular sum, and different graph-labelling conditions (graceful, odd-graceful, edge-magic total, harmonious, and others) become recognisable matrix families. The paper shows that a connected non-tree Topcode-matrix corresponds to at least two Topsnut-gpws, so a single matrix can be published while several non-isomorphic graphs remain usable as private keys. By the union-addition operation, several such matrices merge into one larger matrix, and the paper asserts that splitting that union back into its constituents is hard enough to call the scheme certainly computational security. It then builds every-zero Topcode-matrix groups, graph groups, and number-string groups, and proposes an overall network security mechanism in which each vertex's neighbours must supply group-encrypted permits.

Load-bearing premise

The whole scheme rests on the unproved claim that decomposing a large union Topcode-matrix into its original labelled graphs is computationally infeasible; if that decomposition becomes easy, the public-key and private-key design collapses.

Editorial extensions

If this is right

One published Topcode-matrix can authenticate several private Topsnut-gpws, so a user or community can rotate private keys without changing the public matrix.
A Topcode-matrix can be read out as a number string by the fold-line rules, so graph-based passwords can be stored and transmitted in ordinary text-password fields.
Every-zero Topcode-matrix groups and number-string groups give algebraic operations for encrypting different parts of a dynamic network at different time steps, with any group element usable as zero.
Equivalence results such as Theorem 11 mean a tree's graceful matrix can be converted into odd-graceful, edge-magic-total, or 6C forms, so the same underlying graph can be presented by many matrix shapes.
Hanzi-matrices extend the same framework to Chinese-character codes, so a Chinese sentence can serve as a public key and another as a private key via a linear system.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

My inference: the certainly computational security assertion in Section II.C is only as strong as the splitting problem, and a cheap first check would be to test union matrices against degree-sequence and edge-label recovery algorithms.
My inference: because Theorem 11 identifies matrix families that are equivalent for trees, one could test whether authentication can be made invariant under those transformations, letting a verifier check a canonical matrix class instead of exact private keys.
My inference: the same every-zero group construction could be applied to higher-dimensional arrays or to matrices whose elements are themselves networks, giving hierarchical encryption layers beyond the $3\times q$ case.
My inference: if the fold-line reading rules are made canonical, the generated text strings could be benchmarked against dictionary and entropy attacks to see whether the pictorial structure actually survives in the string form.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my read. The paper's Topcode-matrix is a 3-by-q edge-incidence representation of a labelled graph, and much of what it presents as new—graceful, odd-graceful, edge-magic Topcode-matrices—is a repackaging of standard graph labelings from Gallian's survey. The genuinely useful part is the emphasis on generating text-based passwords from matrices and the worked examples showing how column/row operations and unions of matrices produce password strings. That is a reasonable design direction for the Topsnut-gpw line of work, and the paper has a lot of concrete examples.

The soft spots are serious. The central security assertion, that splitting a large union matrix into its constituent graphs is hard and therefore 'certainly computational security,' is never defined or proved. No adversary model, no reduction, no lower bound. Under the natural reading of the proposed public/private-key split, acceptance is just that the private matrix completes the stored union; then any single column from the public matrix is a valid private key and forgery is trivial. If the intended predicate is to recover the original constituent graphs, the paper's own Fig. 1 shows six non-isomorphic graphs for the same matrix, so the public data do not determine a unique private key. The conclusion's claim that large matrices force attackers to give up is not a derived statement.

The formal mathematics is also shaky. Theorem 5 claims a Topcode-matrix is graphicable iff 2q equals the sum of appearance counts, but because X and Y each have q entries, that condition is an identity. It is not a characterization and cannot filter valid matrices. The Erdős–Gallai theorem is quoted but never applied. The 'every-zero groups' defined by the modular v-operation are indexed copies of cyclic groups; group axioms hold by construction, so they add little. Many other theorems are sketches or assertions. The citation pattern is mostly to the authors' own prior work, which is not damning by itself, but here it means the paper is largely self-referential.

Who gets value from this? Someone working specifically on Topsnut-gpw construction and looking for matrix-based password generation might mine the examples and definitions. As a security paper, it should not be accepted. The load-bearing claim is unsupported, and at least one theorem is false as stated. I would not send it to peer review; I would desk-reject with an invitation to resubmit a version that defines a security game and proves a real characterization.

Referee Report

4 major / 3 minor

Summary. This paper introduces Topcode-matrices, which are 3×q arrays (X,E,Y) interpreted as vertex-edge-vertex encodings of 'Topsnut-gpws,' and it catalogues a large number of restricted families obtained by imposing graph-labelling conditions such as graceful, odd-graceful, edge-magic, and harmonious labelings. The paper defines operations on Topcode-matrices (dual, column/XY exchanging, union-addition, splitting), constructs 'every-zero' matrix groups and number-string groups using modular arithmetic in equations (19)-(20), and proposes text-based passwords and an 'overall security mechanism' for networks in which a large Topcode-matrix union serves as a public authentication and its constituent Topsnut-gpws serve as private keys. The paper also discusses Hanzi-matrices, adjacent ve-value matrices, graph equations, and a list of open questions.

Significance. If the claims were established, the one-to-many correspondence between a Topcode-matrix and non-isomorphic graphs (Fig. 1) could be an interesting source of graphical-password constructions, and the proposed catalog would systematize many graph-labelling notions into a matrix formalism. The paper is also explicit about its open questions, which is a useful feature. However, the load-bearing security statement is not proven, the main graphicability criterion is vacuous, and the group constructions are definitional rather than substantive; the paper contains no formal security model, no reduction, and no computational experiments. The useful parts are the explicit examples and the translation of known graph-labelling conditions into matrix conditions; those do not by themselves establish network security, and the paper provides no machine-checked proofs or reproducibility artifacts.

major comments (4)

[Section II.A.3, Theorem 5] Theorem 5 states that a Topcode-matrix is graphicable if and only if 2q = Σ_{x∈X*}α(x) + Σ_{y∈Y*}α(y). Since X and Y each contain q entries, the right-hand side is identically 2q, so the condition is an identity and cannot discriminate graphicable from non-graphicable matrices. The subsequent citation of the Erdős–Gallai theorem (Theorem 6) is not applied to the degree sequence derived from the appearance counts in Tcode; the degree-sum equation is only necessary, and the Erdős–Gallai inequalities are the missing load-bearing part. As written, Theorem 5 is false as a characterization and invalidates any argument that relies on it to certify that a Topcode-matrix has a graph realization.
[Section II.C and Section V.B.1] The central security claim that Topsnut-gpws are 'certainly computational security' rests on the assertion that splitting a large union Topcode-matrix into its original constituent Topsnut-gpws is computationally difficult. No adversary model, verifier predicate, reduction to a known hard problem, or lower bound is given. Under the natural reading of Section V.B.1, where Tcode = Tcode(Gpub) ⨄ Tcode(Gpri) is the authentication, a presented 'private' matrix is accepted if it completes the stored union; then any single column, such as (7,1,18)^T from the matrix in Eq. (1), is a valid private key and forgery is trivial. If, alternatively, acceptance requires recovering one of the original graphs exactly, Fig. 1 already exhibits six non-isomorphic graphs with the same Topcode-matrix, so the public data do not determine a unique private key. The conclusion that large matrices 'force attackers to give up' is therefore an unsupported assertion rather than a derived security statement.
[Section II.D, Eqs. (19)-(20)] The additive v-operation defines x_{λ,r} = (x_{i,r}+x_{j,r}-x_{k,r}) mod M and λ = i+j-k mod M, so F_m is simply an indexed copy of the cyclic group Z_M acting coordinate-wise on each row. Closure, associativity, the identity (the selected T_k), and inverses hold by construction; the same remark applies to the subtractive operation in Eqs. (25)-(26). Thus the 'every-zero' groups are a notational repackaging of finite cyclic groups, and the paper does not prove any new property of these groups or any connection between the group structure and the hardness of the proposed authentication. This makes the group-theoretic part descriptive rather than a result that can support the security mechanism.
[Section V.A, proof of Theorem 11] The proof of claim (1) of Theorem 11 asserts that in a set-ordered odd-graceful Topcode-matrix 'each x^1_i must be even, and each y^1_i must be odd.' The definition only requires max X < min Y and odd edge labels, and examples with odd-valued X and even-valued Y satisfying both conditions exist (for instance two edges with labels 1 and 3 on X={1,3}, Y={4,4}). Consequently the halving transformation used to recover a set-ordered graceful Topcode-matrix is not well-defined in general, so the claimed equivalence is not established as written.

minor comments (3)

[Throughout] There are numerous typos, including 's ce' in the abstract, 'grapgicable' in Lemma 9, 'T[opcode-matrix' in Section II.C, 'Tosnut-gpw' in Section IV.B, and 'Refereing' in Remark 4; these should be corrected.
[Theorems 1, 2, 4, and 8] Several structural theorems are stated without proof or with only a sketch, including Theorem 1, Theorem 2, Theorem 4, and Theorem 8; the authors should either provide complete proofs or clearly label these statements as conjectures.
[Figures] Several figures (e.g., Figs. 2, 19, and 27) are difficult to read or are not explicitly numbered in the text, which makes the examples harder to verify.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circularity; Theorem 5 is a vacuous graphicability criterion, while the security claim rests on an unproved hardness premise rather than on a circular derivation.

self definitional [Section II.A.3, Theorem 5 (graphicable criterion)]
"Theorem 5. A Topcode-matrix Tcode defined in Definition 4 is graphicable if and only if 2q =∑_{x∈X*} α(x) + ∑_{y∈Y*} α(y), where α(x) (resp. α(y)) is the number of x (resp. y) appeared in X (resp. Y)."

Because X and Y are each q-entry vectors, the right-hand side counts every entry of X and every entry of Y once, so it is identically 2q for every Topcode-matrix. The alleged characterization therefore imposes no restriction; 'graphicable iff true' is a tautology, and the paper immediately needs the Erdős–Gallai inequalities (Theorem 6) to have any real content. The theorem cannot filter graphicable Topcode-matrices or define a hard instance class; at most it restates the definition of a 3×q matrix with an evaluated e-vector.

full rationale

The central security claim ('our Topsnut-gpws are certainly computational security', Section II.C after Fig. 11) is not circular in the sense of a derived prediction; it is an unproved hardness assumption, since no reduction to a known hard problem, adversary game, or verifier predicate is given. This is an evidentiary and correctness gap, not a reduction of the conclusion to its inputs. The every-zero Topcode+-matrix and Topcode−-matrix groups, and the number-string groups, are constructed by the modular formulas (19)-(20) and (25)-(26); their group axioms hold by construction, and no external result is being renamed as a prediction. Theorem 11's equivalences are explicit affine relabelings (for example, x1_i = 2x_i and y1_i = 2y_i − 1), with proofs supplied in the text. Self-citations [11], [12], [14]-[17] introduce the Topsnut-gpw concept but are not load-bearing for the derivations here. The only circular or tautological item found is Theorem 5's vacuous graphicability criterion; it is not used in the security argument, so the paper's central claim has independent, though unsupported, content. Overall circularity is therefore low.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces many named objects, but they are mathematical definitions rather than empirical entities, and no data is fitted. The load-bearing assumptions are standard results (Erdos-Gallai, GB2312 encoding) plus an unproved complexity assumption about splitting Topcode-matrices; the many constants k, d, M are chosen by hand for each naming convention.

free parameters (3)

Modulus M in v-operations = example: 6
The additive and subtractive v-operations (Eqs. 19-20 and 25-26) require choosing a modulus M and a zero element T_k; the group property of the resulting family depends on this choice and is not derived from the matrices themselves.
Constants k, d in (k,d)-Topcode matrices
Introduced by hand in Comp-3, Comp-16, Comp-17 and Parameter-3 through Parameter-6 to define families of matrices; no data determines their values.
Magic constants k, k', k'' in edge-magic and ve-matching conditions
Introduced ad hoc in Cond-14, Cond-15 and Comp-2, Comp-10, Comp-11, Comp-13, Comp-14 to define named matrix classes.

assumptions (4)

standard math Erdos-Gallai degree sequence theorem
Used in Theorem 6 to decide graphicability; the paper cites Bondy and Murty [6] and Harary [4] for sufficiency.
domain assumption GB2312-80 provides a four-digit numeric code for Chinese characters
Used throughout Section IV to construct Hanzi-matrices and Hanzi equations; the encoding is treated as a bijection between characters and strings.
ad hoc to paper Splitting a large union Topcode-matrix into its component Topsnut-gpws is computationally difficult
Implicit in Section II.C and Section V.C as the basis for 'computational security'; no proof or complexity assumption is stated.
ad hoc to paper Degree-sum condition is sufficient for a Topcode-matrix to be graphicable
The paper states this as Theorem 5; it is false without Erdos-Gallai conditions, making it a load-bearing flaw.

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Cite this review

Pith. "Pith review of Topological Coding and Topological Matrices Toward Network Overall Security." pith.science (2026). https://pith.science/paper/ZKYYSFAO

@misc{pith2026190901587,
  author       = {Pith},
  title        = {Pith review of: Topological Coding and Topological Matrices Toward Network Overall Security},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKYYSFAO}},
  note         = {Machine review of arXiv:1909.01587}
}

abstract

A mathematical topology with matrix is a natural representation of a coding relational structure that is found in many fields of the world. Matrices are very important in computation of real applications, s ce matrices are easy saved in computer and run quickly, as well as matrices are convenient to deal with communities of current networks, such as Laplacian matrices, adjacent matrices in graph theory. Motivated from convenient, useful and powerful matrices used in computation and investigation of today's networks, we have introduced Topcode-matrices, which are matrices of order $3\times q$ and differ from popular matrices applied in linear algebra and computer science. Topcode-matrices can use numbers, letters, Chinese characters, sets, graphs, algebraic groups \emph{etc.} as their elements. One important thing is that Topcode-matrices of numbers can derive easily number strings, since number strings are text-based passwords used in information security. Topcode-matrices can be used to describe topological graphic passwords (Topsnut-gpws) used in information security and graph connected properties for solving some problems coming in the investigation of Graph Networks and Graph Neural Networks proposed by GoogleBrain and DeepMind. Our topics, in this article, are: Topsnut-matrices, Topcode-matrices, Hanzi-matrices, adjacency ve-value matrices and pan-Topcode-matrices, and some connections between these Topcode-matrices will be proven. We will discuss algebraic groups obtained from the above matrices, graph groups, graph networking groups and number string groups for encrypting different communities of dynamic networks. The operations and results on our matrices help us to set up our overall security mechanism to protect networks.

Figures

Figures reproduced from arXiv: 1909.01587 by the authors.

**Figure 1.** (a)-(f) are Topsnut-gpws with (splitting) odd-edge-magic colorings and corresponding the Topcode-matrix A, where (a) has no perfect matching, (f) has a perfect matching. We call the matrix A shown in (1) a topological coding matrix (Topcode-matrix) with q = 9 according to the following Definition 4. The above example derives the following questions: Que-1. What properties do Topcode-matrices have? Que-2. Does each … view at source ↗

**Figure 2.** Two Topsnut-gpws (A),(B) and their adjacent matrices. 2) Particular sub-Topcode-matrices: (1) Perfect matching. If we have a subset ES = {ei1 , ei2 , . . . , eim} ⊂ E∗ in a Topcode-matrix Tcode defined in Definition 4, and there exists w ∈ (XY ) ∗ to be not a common end of eij and eit of ES, and the set of all ends of ES is just equal to (XY ) ∗ , then we call ES a perfect matching of Tcode, and (XS ES YS) T a perf… view at source ↗

**Figure 4.** According to six Topsnut-gpws shown in Fig.1, here: (d) A set-ordered splitting harmonious graph (mod 9) and (d-1) a set-ordered harmonious Topcode-matrix (mod 9); (e) a set-ordered splitting edge sum-difference graph and (e-1) a set-ordered edge sum-difference Topcode-matrix; (f) a set-ordered splitting odd-elegant graph (mod 18) and (f-1) a set-ordered odd-elegant Topcode-matrix (mod 18). Comp-2. There exists a co… view at source ↗

Figures from the paper (36 more)

**Figure 3.** Figure 3: According to six Topsnut-gpws shown in Fig.1, here: (a) A set-ordered graceful graph and (a-1) a set-ordered graceful Topcodematrix; (b) a set-ordered splitting odd-graceful graph and (b-1) a setordered odd-graceful Topcode-matrix; (c) a set-ordered splitting edgema…

**Figure 6.** Figure 6: (a) A total graceful Topcode-matrix; (b) a ve-magic total graceful Topcode-matrix; (c) a relaxed edge-magic total Topcodematrix. Parameter-10. A Topcode-matrix Tcode is called an edgemagic graceful Topcode-matrix if it holds Comp-7 and Comp8 true. Parameter-11. A To…

**Figure 5.** Figure 5: (a) A (k, d)-graceful Topcode-matrix; (b) a (k, d)-felicitous Topcode-matrix; (c) a (k, d)-edge-magic total Topcode-matrix; (d) a (k, d)-edge antimagic Topcode-matrix. Parameter-7. A Topcode-matrix Tcode is called a total graceful Topcode-matrix if Comp-3 and Comp-8 ho…

**Figure 8.** Figure 8: An odd-6C-Topcode-matrix B and its dual Topcode-matrix B −1 . Parameter-13. A Topcode-matrix Tcode is called an eedifference odd-edge-magic matching Topcode-matrix if Comp4, Comp-10, Comp-13 and Comp-19 hold true. Parameter-14. A Topcode-matrix Tcode is called an odd…

**Figure 9.** Figure 9: A directed Topsnut-gpw with a directed Topcode-matrix. We show the definition of a directed Topcode-matrix as follows: Definition 5. ∗ A directed Topcode-matrix is defined as −→T code =   x1 x2 · · · xq e1 e2 · · · eq y1 y2 · · · yq   + − =   X −→E Y   + − = [(…

**Figure 10.** Figure 10: A set-intersecting rainbow Topcode-matrix C. (Set-6) We call Tcode defined in Definition 4 a vdistinguishing Topcode-matrix if Nei(xi) 6= Nei(wj ) for any pair of distinct xi , wj ∈ (XY ) ∗ . For each (xi ei yi) T ∈ Tcode, Tcode is called an adjacent v-distinguishing…

**Figure 18.** Figure 18: Fig.18 [PITH_FULL_IMAGE:figures/full_fig_p008_18.png]

**Figure 11.** Figure 11: A Topsnut-gpw G = S4 i=1 Gi is connected, and each Gi admits a graph labelling fi with i ∈ [1, 4]. 1) Popular operations from sets: For a Topcode-matrix Ai = (Xi Ei Yi) with (xi ei yi) T for i ∈ [1, m], and another Topcode-matrix Bj = (Xj Ej Yj ) with (xj ej yj ) T fo…

**Figure 13.** Figure 13: (a) A vertex-coincident graph G(x 0 x 00) obtained by a vertex-coincident operation from (b) to (a); (b) a vertex-split graph G∧x obtained by a vertex-split operation from (a) to (b); (c) an edgecoincident graph G(x 0w 0 x 00w 00) obtained by an edge-coincident opera…

**Figure 12.** Figure 12: A scheme for a half-edge split operation from left to right, and a half-edge coincident operation from right to left. Op-2. A half-edge coincident operation is defined as: Suppose that Nei(x 0 ) ∩ Nei(x 00) = {w}, we coincide x 0 with x 00 into one, denoted as x = (x …

**Figure 14.** Figure 14: (a) An edge-subdivided graph G.w obtained by subdividing a vertex w = (x, y) into an edge xy from (b) to (a); (b) an edgecontracted graph G / xy obtained by contracting an edge xy to a vertex (x, y) from (a) to (b). 11 [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]

**Figure 15.** Figure 15: Two examples for understanding vertex-splitting Topcodematrices. Again we consider do edge-splitting operations on Topcodematrices. In Fig.16, we implement an half-edge-splitting operation to the e = 7 of the Topcode-matrix A shown in (1), the resultant Topcode-mat…

**Figure 17.** Figure 17: An every-zero additive associative Topcode+-matrix group {F6; ⊕} with F6 = {T1, T2, T3, T4, T5, T6} under modular 6. G1 G2 G3 G4 G5 G6 [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]

**Figure 18.** Figure 18: An every-zero graph group {F6(G); ⊕} with F6(G) = {Gi : i ∈ [1, 6]} under modular 6, where each Topsnut-gpw Gi corresponds a Topcode-matrix Ti shown in Fig.17, i ∈ [1, 6]. by the formula below Tb(Ti) =xi,1xi,2xi,3xi,4xi,5ei,5ei,4ei,3ei,2ei,1 yi,1yi,2yi,3yi,4yi,5 (23) …

**Figure 20.** Figure 20: (a) An adjacent TB-paw line; (b) a closed adjacent TB-paw line; (c) a closed TB-paw line, not adjacent. Remark 3. (1) The TB-paw lines shown in Fig.19 can be written into algorithms of polynomial times. (2) TB-paws made from the matrix (27) corresponds a question, cal…

**Figure 19.** Figure 19: Basic rules (also, adjacent TB-paw lines) for producing TBpaws from Topsnut-matrices, Topcode-matrices and Hanzi-GB2312- 80 matrices. Based on the matrix (27), the rule Vo-1 pictured in Fig.19 enables us to write out a TB-paw T e b (M(ai,j )m×n) = a1,1a1,2 · · · a1,n…

**Figure 23.** Figure 23: B1 and B2 are a pair of complementary Topcode-matrices; B1 and B3 are a pair of twin Topcode-matrices; B3 and B4 are a pair of complementary Topcode-matrices. Here, each Topcode-matrix Bi corresponds a graph Gi shown in Fig.22 with i ∈ [1, 4]. 3) Twin odd-graceful Top…

**Figure 21.** Figure 21: A public key Gpub and a private key Gpri can be certified successfully by an authentication G shown in Fig.11, that is, G = Gpub ∪ Gpri, where the set {0, 1, 3, 4, 5, 6, 7, 12, 13, 14} is the common boundaries of Gpub and Gpri. 2) Complementary Topcode-matrices: Let T…

**Figure 24.** Figure 24: Twin odd-graceful Topcode-matrices (Tcode, T k code) with k ∈ [1, 6]. 4) Topcode-matrices of line Topsnut-gpws: We show the line graphs in Fig.25. For i ∈ [1, 6], each line Topsnutgpw Gim is the line graph of the Topsnut-gpw Gi shown in Fig.18. It is not hard to writ…

**Figure 22.** Figure 22: (a) and (b) are a pair of complementary Topsnut-gpws; (c) and (d) are a pair of complementary Topcode-matrices; (a) and (c) are a pair of twin Topsnut-gpws; and G5 = G1 ∪ G2. 2 12 14 12 18 14 18 18 18 18 17 15 13 11 9 7 5 3 1 1 1 5 1 7 5 7 9 11 B     3 264 6 2 …

**Figure 25.** Figure 25: Each line Topsnut-gpw Gim is the line graph of the Topsnutgpw Gi shown in Fig.18 with i ∈ [1, 6]. Condition-2. V (H) = V (H1) ∪ V (H2) and V (H1) ∩ V (H2) = ∅; Condition-3. V (H) = V (H1) ∪ V (H2) and V (H1) ∩ V (H2) 6= ∅; Condition-4. E(H) = E(H1) ∪ E(H2) and E(H1) …

**Figure 26.** Figure 26 [PITH_FULL_IMAGE:figures/full_fig_p017_26.png]

**Figure 28.** Figure 28: A Hanzi H4043 with its Hanzi-gpw H gpw 4043 and two adjacency ve-value matrices A (1)(H4043) and A (2)(H4043) cited from [17]. Tb(22) = 0084083. In this example, we have shown a technique based on adjacent ve-value matrices for producing TBpaws by Hanzis, and it is …

**Figure 29.** Figure 29: A Hanzi-string T = H4476H4734H4662H4311, and a system Y = A(T)X of linear Hanzi equations made by a known Hanzi Hx1x2x3x4 and a unknown Hanzi Hy1y2y3y4 based on [13]. In Fig.30, we have yi = βi,1x1 + βi,2x2 + βi,3x3 + βi,4x4 (mod 10) (36) 18 [PITH_FULL_IMAGE:figures/…

**Figure 30.** Figure 30: Based on the system Y = A(T)X shown in Fig.29, some Hy1y2y3y4 in (b) are unknown in [13]. for i ∈ [1, 4], where (β1,1 β1,2 β1,3 β1,4) = (4 4 4 4), (β2,1 β2,2 β2,3 β2,4) = (4 7 6 3), (β3,1 β3,2 β3,3 β3,4) = (7 3 6 1) and (β4,1 β4,2 β4,3 β4,4) = (6 4 2 1). From Fig.30(a…

**Figure 31.** Figure 31: A matrix equation based on the matrix equation (38). We generalize the matrix equation (38) by setting matrices Y 0 i = (yi,1 yi,2 yi,3 yi,4) T 4×1 , X0 i = (xi,1 xi,2 xi,3 xi,4) T 4×1 and (ai,j )4×4 with 0 ≤ ai,j ≤ 9, such that X4×m = ] m i=1X0 i and Y4×m = ] m i=1Y …

**Figure 32.** Figure 32: Examples for the multiplication and the addition on components of two vectors.              84086538 854248242368 65808440492062002510 82006075 806524852440 cr Ahan 茈？溉条疳躬？汛羧狃崞舁蜂懈跏              5269 6974 3654 62974131 7661 8396 861965535076…

**Figure 34.** Figure 34: Two pure Hanzi matrices from Fig.33. If we have an encoding of Chinese characters, which contains 104 different Chinese characters, then our Hanzi-matrices contain no “?”. E. Matrices with elements to be graphs or Topsnut-gpws 1) Coefficient matrices are number matri…

**Figure 36.** Figure 36: A Topcode-matrix with elements of variables s and i cited from [17]. 4) Topcode-matrices from analytic Hanzis: In Fig.37, a Hanzi H4585 was put into xOy-plane, so we get a new type of matrices, called analytic matrices. 王 (1,2) (0,2) (2,2) (1,0) (2,0) (0,0) (0,1) (1,1…

**Figure 37.** Figure 37: A Hanzi H4585 and its analytic graph, as well as its Topcodematrix with elements to be numbers and coordinates in xOy-plane cited from [17]. 5) Dynamic Topcode-matrices from dynamic networks: Let N (t) be a dynamic network at time step t. Thereby, N (t) has an its ow…

**Figure 38.** Figure 38: Three graph networks (a), (b) and (c) are based on an everyzero Topcode+-matrix group {Ti : i ∈ [1, 6]} shown in Fig.17 by the additive v-operation defined in (19) and (20). Theorem 13. Let {Fp(G), ⊕} be an every-zero ε-group, where Fp(G) = {G1, G2, . . . , Gp}, and …

**Figure 39.** Figure 39: The underlying graph Hunder shown in Fig.38 is labelled with an every-zero graph group shown in Fig.18 to producing a Tosnutgpw G on gray rectangle, and G is equivalent to H shown in Fig.38 (a). {w1, w2, . . . , wd}, where d = deg(w). If a vertex u out of N (t) will …

**Figure 40.** Figure 40: A process of constructing an infinite triangular planar graph admitting a 4-coloring. {G : Avev(G) = Tcode}. Find some connection among the graphs of F(Tcode). Question 11. For a given odd-graceful Topcode-matrix Tcode, find all Topcode-matrices T k code such that (Tc…

**Figure 42.** Figure 42: The rhombus expanded-contracted operation system introduced in [24], where the vertex y is split into y 0 and y 00 from (a) to (b), and the vertex y 0 is split into y1 and y2 from (b) to (c); conversely, identifying two vertices y1 and y2 into one vertex y 0 from (c)…

**Figure 41.** Figure 41: A 4-colorable maximal planar graph G can be contracted into a multiple-edge complete graph K∗ 4 , conversely, K∗ 4 can be v-split into G. The graph W is obtained from G by the 2-degree v-splitting operation, and T obtained from G by a series of v-splitting operations …

**Figure 43.** Figure 43: A maximal triangular planar graph admitting a 4-coloring obtained by the rhombus expanded-contracted operation system introduced in [24]. REFERENCES [1] Xiaogang Wen. Four physical revolutions and the second quantum revolution (talk). The sixth issue of the high-lev…

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Pith tools

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Yujia Li, Chenjie Gu, Thomas Dullien, Oriol Vinyals, Pushmeet Kohli. Graph Matching Networks for Learning the Similarity of Graph Struc- tured Objects. Proceedings of the 36 th International Conference on Machine Learning, Long Beach, USA, 2019. arXiv:1904.12787v1 [cs.LG] 29 Apr 2019

arXiv 2019

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Peter W. Battaglia, Jessica B. Hamrick, Victor Bapst, Alvaro Sanchez- Gonzalez, Vinicius Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, Caglar Gulcehre, Francis Song, Andrew Ballard, Justin Gilmer, George Dahl, Ashish Vaswani, Kelsey Allen, Charles Nash4, Victoria Langston, Chris Dyer, Nicolas Heess, Daan Wierst...

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2008

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work page 2005