Graph Theory Towards Module-K Odd-Elegant Labelling Of Graphical Passwords

. Graph labellings have been applied in many areas of science and engineering, such as in the development of redundant arrays of independent disks which incorporate redundancy utilizing erasure codes, algorithms, design of highly accurate optical gauging systems for use on automatic drilling machines, design of angular synchronization codes, design of optimal component layouts for certain circuit-board geometries, and determining conﬁgurations of simple resistor networks which can be used to supply any of a spec-iﬁed set of resistance values. Based on the idea of of “topological structure plus number theory” and magic type of various labellings for solving network transfer delay by using other type of graphical passwords. We deﬁne a new graph labelling, called Module-K odd-elegant labelling , and ﬁnd some network models that admit our new labelling, and furthermore our methods can be transformed into e ﬀ ective algorithms.


Introduction
No report tells us that much of graphical passwords were applied to business and practice (Ref. [1], [2], [3]). An idea of "topological structure plus number theory" (Topsnut) as an alternative the existing graphical passwords is proposed in [6], which can be realized by graph labellings, and this idea is related with many mathematical conjectures, such as "Every tree is graceful" due to Rosa [4]. The survey article [5] has collected over 1700 papers on research of graph labellings. Motivated from the some graph labellings we define a new labeling, called Module-K odd-elegant labelling. The Module-K odd-elegant labelling can induces some mathematical problems, such as: Every tree admits a Module-K odd-elegant labelling. Furthermore, we can plant this new labelling to other graph labellings, see [7], strongly graceful labelling in [10], odd-graceful labelling in [11], oddelegant labelling in [18], seven other labellings in [13] and so on, which mean that exploring new graphical passwords will bring more new mathematical subjects and new problems. An idea of "topological structure plus number theory" for creating new type of graphical passwords was proposed firstly by Wang et al. in [14] and [15]. In comparing with those graphical passwords mentioned in [16], the new graphical passwords obtained by Topsnut needs less storage and implements quickly in network communication.
G is one with p vertices and q edges. The shorthand symbol [m, n] stands for an integer set {m, m+1, . . . , n}, where m and n are integers with 0 ≤ m < n; the notation [s, t] o indicates an oddset {s, s + 2, . . . , t}, where s and t both are odd integers with 1 ≤ s < t; and the notation [k, ] e indicates an even-set {k, k+2, . . . , }, where k and both are even integers with 0 ≤ k < . Graphical password will be abbreviated as GPW, and "Topsnut-GPW" is the abbreviation of "graphical passwords based on the idea of topological structure plus number theory".
then f is called a graceful labelling, and G is called a graceful graph.
We write the vertex labelling set Suppose that a bipartite graph G admits a graceful labelling f such that max{ f (x) : x ∈ X} < min{ f (y) : y ∈ Y}, where (X, Y) is the bipartition of V(G), we call f a set-ordered graceful labelling, and this case is denoted as f (X) < f (Y) for short ( [17,18]). Definition 2 [9] Suppose that a (p, q)-graph G admits a mapping f : ) and the set of all edge labels is equal to [1, 2q − 1] o , we call f an odd-elegant labelling and G to be odd-elegant. Definition 3 [9] Let (V 1 , V 2 ) be the bipartition of a bipartite (p, q)-graph G. If G admits an oddelegant labelling f such that max{ f (u) : u ∈ V 1 } < min{ f (v) : v ∈ V 2 }, then we call f a set-ordered odd-elegant labelling, and write this case as f (V 1 ) < f (V 2 ). Motivated from the above graph labellings we define a new Module-K odd-elegant labelling as follows.
Definition 4 Suppose that a (p, q)-graph G admits a mapping f : , and the label f (uv) of every edge uv ∈ E(G) is defined as f (uv) = f (u) + f (v) + K(mod 2q) for some given K, and the set of all edge labels is equal to [1, 2q − 1] o or [2, 2q] e , we call f a Module-K odd-elegant labelling and G to be is the bipartition of G, and the Module-K odd- In the current paper, we make the following contributions to the literature on password security: (1) be used conveniently in usually; (2) with strong security, that is, it is difficult to be broken; (3) there are enough graphs and labellings for making desired keys and locks.
For answering the above problems, we prove the series cryptographical graphs have good properties, and show the guarantee for constructing large scale of cryptographical trees from smaller cryptographical trees. The methods used for constructing the desired cryptographical graphs can be transformed into efficient algorithms.

Theorem 1 A connected bipartite graph G is set-ordered graceful if and only if there exists a K such that G admits a Module-K odd-elegant labelling.
Proof. Necessity. Suppose that a bipartite graph G admits a graceful labelling f such that . Thereby, f 1 is a set-ordered graceful labelling of G. A union graphical quasi-group i (G) of order n is defined as: i (G) = {G : V(G) = 1, 2, . . . , n}, the operation between elements of i (G) is the graph union operation " ", and G H produces a simple graph of order n.
(i) (Zero element) the zero element of i (G) is the unique complete graph K n , since G K n = K n .
(ii) (Inverse element) the complementary G of each element G ∈ i (G) is the inverse element of G, since G G = K n .
(iii) Uniqueness and closure. G H is unique and G H ∈ i (G) for any element H ∈ i (G).
(iv) (Associated law) ( A graphical group obtained from the graph Module-K odd-elegant labelling. Let f 1 :  a (p, q) [1, n]) has identical vertex labelling of G k . Each copy of the (p, q)graph G k is written as We call this particular operation as the Module-K odd-elegant additive-sum of graphs.  on a (p, q)graph G 1 forms an Abelian additive group, we call Oele(G) a labelling graphical group based on the graph G hereafter. Furthermore, f (uv) = f (u) + f (v) + K(mod 2q) with some given K.

Proof. First of all, the equation F(T
We are ready to show four principles for a standard group in the following. (1) Zero element. The zero element of Oele(G) is F(T 1 ) as k = 0 in the equation.
We conclude that Oele(G) is a Module-K odd-elegant graphical group. Next, we determine a connection between f 1 (uv) = f 1 (u) + f 1 (v) + K(mod 2q) and f k (u) + f k (v) + K(mod 2q) with some given K in the following. Notice that f 1 (E(T 1 )) = [1, 2q−1] o or [2, 2q] e . Consider the labelled graph T k , take an edge uv ∈ E(T k ), mod 2q), and there are the following cases: The proof of this theorem is complete.  A dual labelling f k of a modular labelling f k is defined by f k (uv) ∈ [1, 2q − 1] o or [2, 2q] e . Let Oele(G) be the set of the graphs H k , where each H k is a labelled Oele(G) by the dual labelling f k of the modular labelling f k with x ∈ V(G) with some given K. We propose two new graphical subject to some con-straint sets as: When G 1 , G 2 , . . . , G m be m bipartite graphs and neither m is odd or even with the Module-K oddelegant labelling g induced by the vertex setlabelling, we show that every simple, connected (p, q)-graph G admits a Module-K odd-elegant labelling, and any tree having a Module-K oddelegant labelling. We propose the following problems for further researching new graphical passwords:

Conclusion and further researches
We are working on new type of graphical passwords based on the idea of "topological structure plus number theory", vividly speaking, new graphical passwords can be called "mathematical fingerprints". And we are exploring transforming human faces into mathematical models in order to recognizing and comparing human faces through mathematical models rather than image recognition among vast of human faces, such works can be considered as "mathematical faces". However, we will make effort to let mathematical fingerprints approximating to human fingerprints. By our experience on working Topsnut-GPWs, we propose the following conjectures: (i) Every bipartite graph admits a Module-K odd-elegant labelling.
(ii) A graph is odd-graceful if and only if it admits a Module-K odd-elegant labelling.
(iii) Any tree T has a Module-K odd-elegant labelling. Some of recent works have demonstrated that textual passwords are particularly vulnerable to targeted online guessing. Textual passwords may be influenced by personal information and textual passwords may be reused, both of which contribute to the vulnerability of textual passwords to online guessing. "Topological structure plus number theory" can be realized by graph labellings, future tests could be designed to answer this question.