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Unitary Addition Cayley Signed Graphs

Journal Name:

  • European Journal of Pure and Applied Mathematics

Publication Year:

  • 2013

Key Words:

Author NameUniversity of Author
Abstract (2. Language): 
A signed graph (or sigraph in short) is an ordered pair S = (Su,), where Su is a graph G = (V, E) and  : E !{+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n, the unitary addition Cayley graph Gn is the graph whose vertex set is Zn, the ring of integers modulo n and if Un denotes set of all units of the ring, then two vertices a and b are adjacent if and only if a + b 2 Un. For a positive integer n, the unitary addition Cayley sigraph n = (u n,) is defined as the sigraph, where u n is the unitary addition Cayley graph and for an edge ab of n, (ab) = ¨ + if a 2 Un or b 2 Un, − otherwise. In this paper, we have obtained a characterization of balanced and clusterable unitary addition Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary addition Cayley sigraphs n, where n has at most two distinct odd prime factors.
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REFERENCES

References: 

[1] B.D. Acharya, A spectral criterion for cycle balance in networks, Journal of Graph Theory,
4(1), 1-11. 1980.
[2] B.D. Acharya, A characterization of consistent marked graphs, National Academy of Science
Letters, 6, 431-440. 1983
[3] M. Acharya, ×-line sigraph of a sigraph, Journal of Combinatorial Mathematics and Combinatorial
Computing, 69, 103-111. 2009.
[4] M. Acharya and D. Sinha, A characterization of sigraphs whose line sigraphs and jump
sigraphs are switching equivalent, Graph Theory Notes N. Y., XLIV, 30-34. 2003.
[5] M. Acharya and D. Sinha, Characterizations of line sigraphs, National Academy of Science
Letters, 28(1 & 2), 31-34. 2005.
[6] M. Acharya and D. Sinha, Common-edge sigraphs, AKCE International Journal of Graphs
and Combinatorics, 3(2), 115-130. 2006.
[7] R. Akhtar, M. Boggess, T. Jackson-Henderson, I. Jiménez, R. Karpman, A. Kinzel and D.
Pritikin, On the unitary Cayley graph of a finite ring, Electronic Journal of Combinatorics,
16(1)(2009), #R117.
[8] N. Alon, Large sets in finite fields are sumsets, Journal of Number Theory, 126(1)(2007),
110-118.
[9] N.D. Beaudrap, On restricted unitary Cayley graphs and symplectic transformations mod-
ulo n, Electronic Journal of Combinatorics, 17(2010), #R69.
[10] M. Behzad and G.T. Chartrand, Line coloring of signed graphs, Elemente der Mathematik,
24(3) (1969), 49-52.
[11] L.W. Beineke and F. Harary, Consistency in marked graphs, Journal of Mathematical Psychology,
18(3)(1978), 260-269.
[12] L.W. Beineke and F. Harary, Consistent graphs with signed points, Rivista di Matematica
per le Scienze Economiche e Sociali, 1(1978), 81-88.
[13] P. Berrizbeitia and R.E. Giudici, On cycles in the sequence of unitary Cayley graphs, Discrete
Mathematics, 282(1-3)(2004), 239-243.
[14] N. Biggs, Algebraic Graph Theory, Second Edition, Cambridge Mathematical Library,
Cambridge University Press, 1993.
[15] M. Boggess, T. Jackson-Henderson, I. Jiménez and R. Karpman, The structure of unitary
Cayley graphs, SUMSRI Journal, (2008), 1-23.
REFERENCES 209
[16] G.T. Chartrand, Graphs as Mathematical Models, Prindle, Weber and Schmidt. Inc.,
Boston, Massachusetts, 1977.
[17] B. Cheyne, V. Gupta and C. Wheeler, Hamilton cycles in addition graphs, Rose-Hulman
Undergraduate Math Journal, 4(1)(2003), 1-17.
[18] F.R.K. Chung, Diameters and eigenvalues, Journal of the American Mathematical Society,
2(2)(1989), 187-196.
[19] J.A. Davis, Clustering and structural balance in graphs, Human Relations, 20(1967), 181-
187.
[20] I.J. Dejter and R.E. Giudici, On unitary Cayley graphs, Journal of Combinatorial Mathematics
and Combinatorial Computing, 18(1995), 121-124.
[21] A. Droll, A classification of Ramanujan unitary Cayley graphs, Electronic Journal of Combinatorics,
17(2010), #N29.
[22] E.D. Fuchs and J. Sinz, Longest induced cycles in Cayley graphs, eprint
arXiv:math/0410308v2 (2004), 1-16.
[23] E.D. Fuchs, Longest induced cycles in circulant graphs, Electronic Journal of Combinatorics,
12(2005), 1-12.
[24] M.K. Gill, Contribution to some topics in graph theory and its applications, Ph.D.
Thesis, Indian Institute of Technology, Bombay, 1983.
[25] C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics,
Springer, 207, 2001.
[26] B.J. Green, Counting sets with small sumset, and the clique number of random Cayley
graphs, Combinatorica, 25(2005), 307-326.
[27] D. Grynkiewicz, V.F. Lev and O. Serra, The connectivity of addition Cayley graphs, Electronic
Notes in Discrete Mathematics, 29(2007), 135-139.
[28] D. Grynkiewicz, V.F. Lev and O. Serra, Connectivity of addition Cayley graphs, Journal of
Combinatorial Theory, Series B, 99(1)(2009), 202-217.
[29] F. Harary, On the notion of balance of a signed graph, Michigan Mathematical Journal,
2(1953), 143-146.
[30] F. Harary, Graph Theory, Addison-Wesley Publishing Company, Reading, Massachusetts,
1969.
[31] F. Harary and J.A. Kabell, A simple algorithm to detect balance in signed graphs, Mathematical
Social Sciences, 1(1980-81), 131-136.
REFERENCES 210
[32] F. Harary and J.A. Kabell, Counting balanced signed graphs using marked graphs, Proceedings
of the Edinburgh Mathematical Society, 24(2)(1981), 99-104.
[33] C. Hoede, A characterization of consistent marked graphs, Journal of Graph Theory,
16(1)(1992), 17-23.
[34] W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electronic Journal of
Combinatorics, 14(2007), #R45.
[35] V.F. Lev, Sums and differences along Hamiltonian cycles, Electronic Notes in Discrete Mathematics,
28(2007), 25-31.
[36] V.F. Lev, Sums and differences along Hamiltonian cycles, Discrete Mathematics,
310(3)(2010), 575-584.
[37] H.N. Ramaswamy and C.R. Veena, On the energy of unitary Cayley graphs, Electronic
Journal of Combinatorics, 16(2009), #N24.
[38] E. Sampathkumar and S.B. Chikkodimath, Semitotal graphs of a graph-I, Journal of Karnatak
University Science, 18(1973), 274-280.
[39] T. Sander, Eigenspaces of Hamming graphs and unitary Cayley graphs, Ars Mathematica
Contemporanea, 3(2010), 13-19.
[40] D. Sinha, New frontiers in the theory of signed graphs, Ph.D. Thesis, University of
Delhi, 2005.
[41] D. Sinha and A. Dhama, Sign-Compatibility of some derived signed graphs, Indian Journal
of Mathematics, 55(1), 2013.
[42] D. Sinha and P. Garg, On the unitary Cayley signed graphs, Electronic Journal of Combinatorics,
18(2011), #P229.
[43] D. Sinha, P. Garg and A. Singh, Some properties of unitary addition Cayley graphs, Notes
on Number Theory and Discrete Mathematics, 17(3)(2011), 49-59.
[44] D. Sinha and P. Garg, Some results on semi-total signed graphs, Discussiones Mathematicae.
Graph Theory, 31(4)(2011b), 625-638.
[45] D.B. West, Introduction to Graph Theory, Prentice-Hall of India Pvt. Ltd., 1996.
[46] T. Zaslavsky, Glossary of signed and gain graphs and allied areas, II Edition, Electronic
Journal of Combinatorics, #DS8(1998).
[47] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas, VII
Edition, Electronic Journal of Combinatorics, #DS8(1998). (Latest update: 8th Edition,
September 2012).
[48] T. Zaslavsky, Signed analogs of bipartite graphs, Discrete Mathematics, 179(1998), 205-
216, (1998).

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