Characterization of 2-Path Product Signed Graphs with Its Properties

Sinha, Deepa; Sharma, Deepakshi

doi:https://doi.org/10.1155/2017/1235715

Computational Intelligence and Neuroscience

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Abstract Introduction Conclusion Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 1235715 | https://doi.org/10.1155/2017/1235715

Characterization of 2-Path Product Signed Graphs with Its Properties

Deepa Sinha¹and Deepakshi Sharma¹

Academic Editor: Silvia Conforto

Received09 Mar 2017

Accepted22 May 2017

Published06 Jul 2017

Abstract

A signed graph is a simple graph where each edge receives a sign positive or negative. Such graphs are mainly used in social sciences where individuals represent vertices friendly relation between them as a positive edge and enmity as a negative edge. In signed graphs, we define these relationships (edges) as of friendship (“” edge) or hostility (“” edge). A 2-path product signed graph of a signed graph is defined as follows: the vertex set is the same as and two vertices are adjacent if and only if there exists a path of length two between them in . The sign of an edge is the product of marks of vertices in where the mark of vertex in is the product of signs of all edges incident to the vertex. In this paper, we give a characterization of 2-path product signed graphs. Also, some other properties such as sign-compatibility and canonically-sign-compatibility of 2-path product signed graphs are discussed along with isomorphism and switching equivalence of this signed graph with 2-path signed graph.

1. Introduction

Signed graph forms one of the most vibrant areas of research in graph theory and network analysis due to its link with behavioural and social sciences. The earliest appearance of signed graphs can be traced back to Heider [1] and Cartwright [2]. From that time to recently, signed theory has evolved rapidly with signed graphs being linked to algebra [3–5], social networks [6, 7], other models [8, 9], and graph spectra [10] to name few. In graph theory, itself signed graphs have been used to define many properties and new concepts. In [11, 12] the signed graph of line signed graphs is discussed, whereas [13, 14] talks about common edge signed graphs. The work in [15, 16] generalises the graceful graphs to signed graphs. The colouring of signed graphs is reported in [17–19]. The connection between the intersection graphs of neighborhood and signed graphs has also been studied [20–24]. Recently a Coxeter spectral analysis and a Coxeter spectral classification of the class of edge-bipartite graphs (that is a class of signed (multi)graphs) is developed in the papers [25–27] in relation to Lie theory problems, quasi Cartan matrices, Dynkin diagrams, Hilbert’s X Problem, combinatorics of Coxeter groups, and the Auslander-Reiten theory of module categories and their derived categories. In this paper, we were mainly driven to carry out work in the area of signed graphs derived from 2-path product operations, which primarily deals with the structural reconfiguration of the structure of dynamical systems under prescribed rules and the rules are designed to address a variety of interconnections among the elements of the system. We have obtained some theoretical results (some of which are presented in [28]) with a hope of building necessary conceptual resources for applications. For standard terminology and notation in graph theory one can refer to Harary [29] and West [30] and for signed graph literature one can read Zaslavsky [19, 31, 32]. Throughout the text, we consider finite, undirected graph with no loops or multiple edges.

A signed graph is an ordered pair , where is a graph , called the underlying graph of and is a function from the edge set of into the set , called the signature (or in short) of . Alternatively, the signed graph can be written as , with , , and in the above sense. A signed graph is all-positive (resp., all negative) if all its edges are positive (negative); further, it is said to be homogeneous if it is either all-positive or all negative and heterogeneous otherwise. The positive (negative) degree of a vertex denoted by is the number of positive (negative) edges incident on the vertex and . The negation of a signed graph is obtained by reversing the sign of edges of . Let be an arbitrary vertex of a graph . We denote the set consisting of all the vertices of adjacent to by . This set is called the of and sometimes we call it as of . A is an ordered pair where is a signed graph and is a function from the vertex set of into the set , called a of . denotes the set of all markings on vertices of . For any vertex is called canonical marking. The marking on the vertices will be specified in the whole text as the case may be.

, , and . A vertex with a marking is denoted by . A cycle in a signed graph is said to be positive if the product of the signs of its edges is positive or, equivalently, if the number of negative edges in it is even. A cycle which is not positive is said to be negative.

A signed graph is line balanced or balanced if all its cycles are positive. The partition criterion to characterize the balance property of a signed graph is given by Harary. A marked graph is vertex or point balanced if it does not contain odd number of negative vertices. A signed graph is sign-compatible [35] if there exists a marking of its vertices such that the end vertices of every negative edge receive “−” marks in and no positive edge in has both of its ends assigned “−” mark in ; it is sign-incompatible otherwise. A canonically marked graph is said to be canonically sign-compatible (or C-sign-compatible) if end vertices of every negative edge receive “−” sign and no positive edge has both of its ends assigned “−” under .

The idea of switching a signed graph was introduced by Abelson and Rosenberg [36] in connection with structural analysis of social behaviour and may be formally stated as follows: given a marking of a signed graph , switching with respect to is the operation of changing the sign of every edge of to its opposite whenever its end vertices are of opposite signs in (also see Gill and Patwardhan [37, 38]). The signed graph obtained in this way is denoted by and is called the -switched signed graph or just switched signed graph when the marking is clear from the context. Further, a signed graph switches to signed graph (or that they are switching equivalent to each other), written as , whenever there exists such that , where “” denotes the isomorphism between any two signed graphs in the standard sense. Two signed graphs and are cycle isomorphic if there exists an isomorphism , where and are underlying graph of and , respectively, such that the sign of every cycle in equals the sign of in .

Assume that is a signed graph. We associate with the 2-path signed graph [39] defined as follows: the vertex set is same as the original signed graph and two vertices , are adjacent if and only if there exists a path of length two in . The edge is negative if and only if all the edges in all the two paths in are negative otherwise the edge is positive (see Figure 1). The 2-path product signed graph [40] is defined as follows: The vertex set is same as the original signed graph and two vertices , are adjacent if and only if there exists a path of length two in . The sign is canonical marking (see Figure 2).

Property 1 (see [39]). A 2-subset in a neighborhood of a vertex in a given signed graph has property if for some and for each containing , .

In the first section, we give a characterization of 2-path product signed graph, followed by a theorem of finding the degree of each vertex in . Also, we find when a 2-path product graph is isomorphic and switching equivalent to its negation. Next, we find when is all negative for a given . The following two sections are dedicated to signed graph properties sign-compatibility and canonical-sign-compatibility. The last section deals with the isomorphism and switching equivalence of the two types of 2-path graphs of signed graphs.

2. Characterization of 2-Path Product Signed Graph

We require the following theorems for the characterization of 2-path product signed graph.

Theorem 2 (see [41]). A signed graph is vertex balanced if and only if it is possible to assign signs to the edges of such that the mark of any vertex is equal to the product of the signs of the edges incident to .

The following characterization of 2-path graphs was given by Acharya and Vartak.

Theorem 3 (see [42]). A connected graph with vertices is of the 2-path graph form , with some graph if and only if contains a collection of complete subgraphs such that for each (i);(ii);(iii) and there exists containing .

Theorem 4 (see [39]). A connected sigraph with vertices is a 2-path sigraph of some sigraph if and only if contains a collection of complete subsigraphs with marked vertices such that, for each , the following hold:(i);(ii);(iii) with sign ; then there exists containing where and if then is a pair in .

The following proposition is evident from [43, 44].

Proposition 5. 2-path product signed graph of a signed graph is always balanced.

We give a characterization for 2-path product signed graph.

Theorem 6. A connected signed graph with vertices is of the 2-path product signed graph form with some signed graph if and only if the underlying graph is a 2-path graph and is both line balanced and vertex balanced.

Proof. Necessity. Suppose is of the 2-path product signed graph form with vertices . Now from Theorem 3, there exist complete subsigned graphs such that (i), (ii), and (iii) hold. Let us consider the set of neighborhood of a vertex in . For each vertex in there is a neighborhood , hence such subsets of neighborhoods. Clearly since we consider open neighborhood, , also if a vertex , then is an edge in and hence . And if is an edge in then and are adjacent to a vertex in . That is such that since each vertex has a marking in . We know that is a canonically marked signed graph; thus each vertex has a marking . Now let be the neighborhood of a vertex with marked vertices retaining the marking from . Then clearly since all three properties (i), (ii), and (iii) of Theorem 3 are satisfied and also by Theorem 2, and Proposition 5, is line balanced and vertex balanced.
Sufficiency. Let be a given signed graph such that its underlying graph is a 2-path graph and is both line balanced and vertex balanced. Then by Theorem 3, it can be written as the union of complete subsigned graphs of marked vertices such that for each , (i), (ii), and (iii) hold. Now associate a vertex to and join to all the vertices in and giving the edge sign as that of the product of marking on and where . Let the signed graph thus obtained be . Next we show that . Obviously , where and are underlying graph of and , respectively. Let be an edge with the sign ; then , where and are markings on and , respectively. By hypothesis, for some . Hence we will associate a vertex to and let its marking be . By definition, the sign of edge in is . That is . Therefore, is the signed graph such that .

The characterization of 2-path signed graph in Theorem 4 provides us with a mechanism to check if a given signed graph is 2-path of some signed graph, which is discussed in Algorithm 1. This has been rigorously studied elsewhere in the author’s contribution which is fully devoted to 2-path signed graphs and its properties. Thus Algorithm 2 using Algorithm 1 detects if the given signed graph is 2-path product signed graph and find the original signed graph. In Algorithm 2, we use the adjacency matrix and its order to find the original signed graph. Algorithm 3 is used to find the 2-path product signed graph for a given signed graph.

Input. The adjacency matrix
Output. If is a 2-path for some signed graph then returns .
Process
(1) Collect all the cliques for each vertex , using Bron-Kerbosch algorithm [33].
(2) Mark every vertex by + and then − in each clique.
(3) Calculate , which represent the all possible combinations generated by each marked vertex from the
clique.
(4) for to do
(5) Select
(6) for to do
(7) Select
(8) if then
(9)
(10) for to do
(11) for to do
(12) if then
(13)
(14) else
(15) if then
(16)
(17) else
(18)
(19)
(20) for to do
(21) for to do
(22) if then
(23) go to (25)
(24) else
(25) go to (23)
(26) For all the combinations of elementary swamping operations on either rows or columns in , repeat (4).
(27) If all the combinations are checked and no such matrix is obtained then no such graph exist.
(28) If such a signed graph exist then is the adjacency matrix of required signed graph whose 2-path signed
graph is .

Input. The adjacency matrix of signed graph and dimension
Output. If is a 2-path for some signed graph then returns its adjacency matrix .
Process
(1) We use Algorithm 1 to detect if is a 2-path signed graph.
(2) Use algorithm in [34] to check if is balanced.
(3) for to do
(4) ; for to do
(5) if then
(6) ;
(7) ;
(8) for to do
(9)
(10) if then
(11) The given signed graph is not a 2-path product signed graph
(12) else
(13) The given signed graph is a 2-path product signed graph

Input. Adjacency matrix and dimension .
Output. Adjacency matrix of 2-path signed graph
Process
(1) Enter the order and adjacency matrix of for a given signed graph .
(2) Collect all the pairs for given signed graph .
(3) for to do
(4) for to do
(5) if then
(6) ;
(7) for to do
(8) for to do
(9) for to do
(10) if then
(11) if then
(12)
(13)
(14) else
(15)
(16)

Theorem 7. If being the canonical marking of a vertex , then the degree of the vertex in , for a given signed graph , is given by the following:(i)If then positive degree of in and the negative degree of .(ii)If then positive degree of in and the negative degree is .

Proof. By Theorem 6 the neighborhoods of a vertex of gives the edges in . That is, if for some , then is an edge in . Thus gives the number of vertices which form an edge with in . And since the marking is canonical in thus positive edges in are given by vertices with same marking. Thus a vertex in is given by the following:(i)If then positive degree of in and the negative degree of .(ii)If then positive degree of in and the negative degree is .

Theorem 8. , if and only if is a signed graph with each vertex of even degree.

Proof.
Necessity. Let ; then clearly the underlying graph of is such that . Also since is a canonically marked signed graph with each vertex of even degree, the mark on every vertex will be the product of edges incident to it. Let if possible be a vertex with number of positive edges incident to and be the number of negative edges incident to it. Then one of the following cases arises.
Case 1. Let be even; then is also even since the total number of edges incident to is even. In negation of , will again be even (since is even in ). Thus both retain the same marking for .
Case 2. Let be odd then is odd. Clearly ; also in is again negative. Thus in both and the marking of is −.
Clearly, since marking on each vertex remains the same so their 2-path product signed graphs remain isomorphic.
Sufficiency. Let . Let if possible be a vertex with odd degree. Let be the number of positive edges incident to and be the negative edges incident to ; then the following cases arise: (i)If is odd then is even. Consequently, receives a positive marking in , but in its negation the number of negative edges becomes odd and hence the sign is reversed.(ii)If is even then is odd. The marking in and is again reversed.Thus if the signed graph has odd degree vertices then the 2-path product graphs of and are not isomorphic, which is a contradiction.

Corollary 9. For any signed graph , .

Proof. Clearly, , where is underlying graph of . Next we know that is always balanced, for every signed graph . Thus all cycles are positive and have even number of negative edges. Thus both and will possess cycles with even number of negative edges. Thus .

Theorem 10. A 2-path product signed graph of a given signed graph is all negative if and only if is either a cycle of length or a signed path and does not contain a subsigned path or , in where .

Proof.
Necessity. Let for a given its 2-path product signed graph be all negative. Clearly, the signed graph can be a tree or a cycle. Now if is not a cycle or tree then will consist of cliques which can not be all negative since cliques always consist of a cycle of length three which can never be all negative as 2-path product signed graphs are always balanced. Clearly, 2-path graph of a cycle of odd length is self-isomorphic. Thus the cycle of odd length can not generate all negative 2-path product graphs. The 2-path graphs of cycles of even length say are disjoint cycles of length each. So if is odd then also the 2-path product signed graph can never be all negative. Thus, a cycle of length can generate all negative 2-path product signed graphs. To produce all negative 2-path product signed graph , can not have subsigned path or , on any subsigned path since then will be a positive edge in .
Also if there is a tree with a vertex of degree greater than two, then clearly it gives rise to a clique containing cycles of length three in , thus having at least one positive edge. Hence the tree can not have a vertex of degree greater than two. Thus, it is a signed path.
Sufficiency. let is either a cycle of length or a signed path and does not contain a subsigned path or , where . Clearly will be disjoint cycles in case of cycle except for where it will be two disjoint signed paths. And in case of signed path will be disjoint paths. And since always for any subsigned path in , , and will occupy opposite mark in , thus it makes edge negative in . Thus is all negative.

3. Sign-Compatibility of 2-Path Product Signed Graphs

In this section, we give a characterization of sign-compatibility for 2-path product signed graphs.

Theorem 11 (see [35]). A signed graph is sign-compatible if and only if does not contain a subsigned graph isomorphic to either of the two signed graphs in Figure 3, formed by taking the path with both the edges and negative and the edge positive, and formed by taking and identifying the vertices and .

Theorem 12. A 2-path product signed graph of a signed graph is sign-compatible if and only if (i) does not contain a heterogeneous canonically marked triangle or ;(ii) does not consist of the canonically marked subsigned path or , where .

Proof.
Necessity. Let 2-path product signed graph of a signed graph be sign-compatible. To prove (i) and (ii), let consist of a heterogeneous marked triangle ; then there exist two vertices with same mark and one vertex with different mark. Clearly the 2-path product signed graph will contain triangle with two negative edges and one positive edge. Thus will not be sign-compatible, which is a contradiction. Again if contains a heterogeneous canonically marked then will consist of a forbidden triangle in Figure 3. Hence (i) holds. Let if possible consist of the canonically marked subsigned path or , where . Then will contain a forbidden in Figure 3; thus will not be sign-compatible which is a contradiction to our assumption. Hence (ii) holds.
Sufficiency. Let (i) and (ii) hold. To show is sign-compatible, let if possible not be sign-compatible. Then must consist of subsigned graph isomorphic to Figure 3, which is not possible as then either (i) or (ii) does not hold true. Hence is sign-compatible.

4. C-Sign-Compatibility of 2-Path Product Signed Graphs

This section gives the C-sign-compatibility of 2-path product signed graphs.

Proposition 13 (see [45]). Every C-sign-compatible signed graph is sign-compatible.

Theorem 14 (see [45]). A signed graph , is C-sign-compatible if and only if the following holds for : (i)For every vertex either or and(ii)For every positive edge in either or .

Theorem 15. A 2-path product signed graph of a signed graph is C-sign-compatible if and only if (i) is sign-compatible;(ii) does not contain a subsigned path , of vertices where ;(iii)if there exist a subsigned path of vertices in ; then either or , where ;

Proof.
Necessity. Let be C-sign-compatible then clearly it is sign-compatible by Proposition 13. Let us suppose contains a subsigned graph ; then clearly is a positive edge in such that and , which is a contradiction to the fact that is C-sign-compatible. Hence does not contain subsigned path .
Let there exist a subsigned path on vertices in , such that and . Then is a positive edge in with both the vertices having negative degrees which is a contradiction to Theorem 14. Thus (i), (ii), and (iii) hold.
Sufficiency. Let (i), (ii), and (iii) hold. Then clearly for each positive edge in either or . Hence by Theorem 14, is C-sign-compatible.

5. Isomorphism and Switching Equivalence of and

In this section, we give the switching equivalent and isomorphism for the two definitions of 2-path signed graphs.

Theorem 16 (see [46]). Given a graph , any two signed graphs are switching equivalent if and only if they are cycle isomorphic.

Theorem 17 (see [39]). For a signed graph of order n, its 2-path signed graph is balanced if and only if for all sequences of vertices in such that for some ; then the pairs having property are even in each sequence.

Theorem 18. The 2-path signed graph and 2-path product graph are switching equivalent if and only if is balanced.

Proof.
Necessity. if and are switching equivalent then they are cycle isomorphic and hence is balanced.
Sufficiency. Clearly, . Next, we know that is always balanced. For balanced , each cycle of and will be positive which implies that and will be cycle isomorphic. Thus, by Theorem 16, and are switching equivalent.

Theorem 19. The 2-path signed graph and 2-path product graph are isomorphic, if and only if there exists subsigned path or in ; then , satisfies property.

Proof.
Necessity. For a signed graph , let its 2-path signed graph and 2-path product graph be isomorphic; here if is a negative (positive) in then it is negative in . All the pair of vertices are negative in and have property P. If there exist subsigned path and where in then is a negative edge in and thus satisfies property P.
Sufficiency. Let there exist subsigned path and in then has property P. To show 2-path signed graph and 2-path product graph are isomorphic. Clearly , being the underlying graph of . Thus we need to show that the sign convention remains the same in and . This is true since the end vertices of every negative edge of have property and hence is a negative edge in . And thus 2-path signed graph and 2-path product graph are isomorphic.

6. Conclusion

In this paper, we have worked on 2-path product signed graph of a given signed graph . A 2-path product signed graph is the signed graph where the vertex set is same as the original signed graph and two vertices are adjacent if and only if there exists a path of length two in . The sign being canonical marking. We give its algorithmic characterization along with its properties like sign-compatibility and C-sign-compatibility. Also, we find the isomorphism of 2-path product signed graph and its negation. We next find isomorphism of 2-path signed graph and 2-path product signed graphs.

Conflicts of Interest

All the authors declare that they have no conflicts of interest regarding publication of this paper.

Acknowledgments

The authors wish to thank Professor Thomas Zaslavsky, Binghamton University, State University of New York, for going through the paper and giving suggestions. His input to this paper has helped the authors to bring the paper in the present form.

References

F. Heider, “Attitudes and Cognitive Organization,” Journal of Psychology: Interdisciplinary and Applied, vol. 21, no. 1, pp. 107–112, 1946.
View at: Publisher Site | Google Scholar
D. Cartwright and F. Harary, “Structural balance: a generalization of Heider's theory,” Psychological Review, vol. 63, no. 5, pp. 277–293, 1956.
View at: Publisher Site | Google Scholar
T. Zaslavsky, Matrices in the theory of signed simple graphs. https://arxiv.org/abs/1303.3083.
P. Sharma and M. Acharya, “Balanced signed total graphs of commutative rings,” Graphs and Combinatorics, vol. 32, no. 4, pp. 1585–1597, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
D. Sinha, D. Ayushi, and B. D. Acharya, “Unitary addition cayley signed graphs,” European Journal of Pure and Applied Mathematics, vol. 6, no. 2, pp. 189–210, 2013.
View at: Google Scholar
J. Kunegis, Applications of structural balance in signed social networks. https://arxiv.org/abs/1402.6865.
J. Leskovec, D. P. Huttenlocher, and J. M. Kleinberg, “Signed networks in social media,” in Proceedings of the 28th International Conference on Human Factors in Computing Systems (CHI '10), pp. 1361–1370, Atlanta, Ga, USA, April 2010.
View at: Publisher Site | Google Scholar
W. Kager, M. Lis, and R. Meester, “The signed loop approach to the Ising model: foundations and critical point,” Journal of Statistical Physics, vol. 152, no. 2, pp. 353–387, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
L. Ou-Yang, D.-Q. Dai, and X.-F. Zhang, “Detecting Protein Complexes from Signed Protein-Protein Interaction Networks,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 12, no. 6, pp. 1333–1344, 2015.
View at: Publisher Site | Google Scholar
D. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, New York, NY, USA, 2010.
View at: MathSciNet
D. Sinha and P. Garg, “Canonical consistency of semi-total line signed graphs,” National Academy Science Letters, vol. 38, no. 5, pp. 429–432, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
K. A. Germina, S. Hameed K., and T. Zaslavsky, “On products and line graphs of signed graphs, their eigenvalues and energy,” Linear Algebra and its Applications, vol. 435, no. 10, pp. 2432–2450, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
M. Acharya and D. Sinha, “Common-edge sigraphs,” AKCE International Journal of Graphs and Combinatorics, vol. 3, no. 2, pp. 115–130, 2006.
View at: Google Scholar | MathSciNet
P. S. Reddy, E. Sampathkumar, and M. S. Subramanya, “Common-edge signed graph of a signed graph,” Journal of the Indonesian Mathematical Society, vol. 16, no. 2, pp. 105–113, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
M. Acharya and T. Singh, “Graceful signed graphs,” Czechoslovak Mathematical Journal, vol. 54, no. 2, pp. 291–302, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
M. Acharya and T. Singh, “Graceful signed graphs: Ii. the case of signed cycles with connected negative sections,” Czechoslovak Mathematical Journal, vol. 55, no. 1, pp. 25–40, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
M. Behzad and G. Chartrand, “Line-coloring of signed graphs,” Elemente der Mathematik, vol. 24, pp. 49–52, 1969.
View at: Google Scholar | MathSciNet
T. Fleiner and G. Wiener, “Coloring signed graphs using dfs,” Optimization Letters, vol. 10, no. 4, pp. 865–869, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
T. Zaslavsky, “Signed graphs,” Discrete Applied Mathematics, vol. 4, no. 1, pp. 47–74, 1982.
View at: Publisher Site | Google Scholar | MathSciNet
R. Rangarajan, M. S. Subramanya, and P. S. Reddy, “Neighborhood signed graphs,” Southeast Asian Bulletin of Mathematics, vol. 36, no. 3, pp. 389–397, 2012.
View at: Google Scholar | MathSciNet
D. Sinha and A. Dhama, “Negation switching invariant signed graphs,” Electronic Journal of Graph Theory and Applications. EJGTA, vol. 2, no. 1, pp. 32–41, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
P. S. K. Reddy, P. N. Samanta, and P. S. Kavita, “On signed graphs whose two path signed graphs are switching equivalent to their jump signed graphs,” Mathematical Combinatorics, vol. 1, pp. 74–79, 2015.
View at: Google Scholar
D. Sinha and D. Sharma, “Algorithmic characterization of signed graphs whose two path signed graphs and square graphs are isomorphic,” in Proceedings of the 2014 International Conference on Soft Computing Techniques for Engineering and Technology, ICSCTET 2014, IEEE, pp. 1–5, August 2014.
View at: Publisher Site | Google Scholar
D. Sinha and D. Sharma, “On Square and 2-path Signed Graph,” Journal of Interconnection Networks, vol. 16, no. 1, Article ID 1550011, 2016.
View at: Publisher Site | Google Scholar
D. Simson, “Symbolic algorithms computing GRAm congruences in the Coxeter spectral classification of edge-bipartite graphs, I. A GRAm classfication,” Fundamenta Informaticae, vol. 145, no. 1, pp. 19–48, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
A. Mroz, “Congruences of edge-bipartite graphs with applications to Grothendieck group recognition I. inflation algorithm revisited,” Fundamenta INFormaticae, vol. 146, no. 2, pp. 121–144, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
A. Mroz, “Congruences of edge-bipartite graphs with applications to grothendieck group recognition ii. coxeter type study,” Fundamenta Informaticae, vol. 146, no. 2, pp. 145–177, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
D. Sinha and S. Deepakshi, “2-path product signed graph,” in Proceedings of the National Conference on Algebra, Analysis, Coding and Cryptography, October 2016.
View at: Google Scholar
F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, USA, 1969.
View at: MathSciNet
D. B. West, Introduction to Graph Theory, Prentice hall Upper, Saddle River, NJ, USA, 2001.
T. Zaslavsky, “Signed analogs of bipartite graphs,” Discrete Mathematics, vol. 179, no. 1-3, pp. 205–216, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
T. Zaslavsky, “A mathematical bibliography of signed and gain graphs and allied areas,” Electronic Journal of Combinatorics, vol. 1000:DS8, 2012.
View at: Google Scholar | MathSciNet
C. Bron and J. Kerbosch, “Algorithm 457: finding all cliques of an undirected graph,” Communications of the ACM, vol. 16, no. 9, pp. 575–577, 1973.
View at: Publisher Site | Google Scholar
F. Harary and J. A. Kabell, “A simple algorithm to detect balance in signed graphs,” Mathematical Social Sciences, vol. 1, no. 1, pp. 131–136, 1980.
View at: Publisher Site | Google Scholar | MathSciNet
D. Sinha and Ai. Dhama, “Sign-compatibility of some derived signed graphs,” Mapana-Journal of Sciences, vol. 11, no. 4, p. 14, 2012.
View at: Google Scholar
R. P. Abelson and M. J. Rosenberg, “Symbolic psycho‐logic: A model of attitudinal cognition,” Behavioral Science, vol. 3, no. 1, pp. 1–13, 1958.
View at: Publisher Site | Google Scholar
M. K. Gill and G. A. Patwardhan, “Switching invariant two-path signed graphs,” Discrete Mathematics, vol. 61, no. 2-3, pp. 189–196, 1986.
View at: Publisher Site | Google Scholar | MathSciNet
M. K. Gill and G. A. Patwardhan, “A characterization of sigraphs which are switching equivalent to their line sigraphs,” Journal of Mathematical and Physical Sciences, vol. 15, no. 6, pp. 567–571, 1981.
View at: Google Scholar | MathSciNet
D. Sinha and D. Sharma, “On 2-path signed graphs,” in Proceedings of the 2016 International Workshop on Computational Intelligence (IWCI), IEEE, pp. 218–220, Dhaka, Bangladesh, December 2016.
View at: Publisher Site | Google Scholar
P. S. K. Reddy and M. S. Subramanya, “Note on path signed graphs,” Notes on Number Theory and Discrete Mathematics, vol. 4, no. 16, p. 1, 2009.
View at: Google Scholar
E. Sampathkumar, “Point signed and line signed graphs,” National Academy Science Letters, vol. 7, no. 3, pp. 91–93, 1984.
View at: Google Scholar | MathSciNet
A. B. Devadas and M. N. Vartak, “Open neighborhood graphs,” Tech. Rep. 7, Indian Institute of Tech- nology Department of Mathematics Research Report, 1973.
View at: Google Scholar
J. A. Davis, “Clustering and structural balance in graphs,” Social networks. A developing paradigm, pp. 27–34, 1977.
View at: Google Scholar
F. Harary, “On the notion of balance of a signed graph,” The Michigan Mathematical Journal, vol. 2, no. 2, pp. 143–146, 1953.
View at: Publisher Site | Google Scholar | MathSciNet
D. Sinha and A. Dhama, “Canonical-sign-compatibility of some signed graphs,” Journal of Combinatorics Information & System Sciences, vol. 38, no. 1-4, 129 pages, 2013.
View at: Google Scholar
T. Sozanśki, “Enumeration of weak isomorphism classes of signed graphs,” Journal of Graph Theory, vol. 4, no. 2, pp. 127–144, 1980.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2017 Deepa Sinha and Deepakshi Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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