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Signatures and signed switching classes

Peter J Cameron, Albert L Wells
1986 Journal of combinatorial theory. Series B (Print)  
In this paper an analogous theory of signed switching classes and signatures is developed using similar techniques.  ...  The reader is advised to consult Wells [ 71 and Zaslavsky [ 111 for other approaches to signed switching classes and signatures.  ... 
doi:10.1016/0095-8956(86)90088-2 fatcat:je7hqbfjknbllbru5yd3qywlh4

Eulerian partitions for configurations of skew lines [article]

Roland Bacher, David Garber
2010 arXiv   pre-print
We also describe an algorithm which constructs a spindle-permutation for a given switching class, or proves non-existence of such a spindle-permutation.  ...  In this paper, which is a complement of BG, we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators.  ...  Acknowledgments The second author wishes to thank the Institut Fourier where most of this work was done and Mikhail Zaidenberg for hosting his stay.  ... 
arXiv:1006.3447v1 fatcat:zyat5ohj3bcnhlpnhxnx3agjdi

Signed Complete Graphs on Six Vertices [article]

Deepak, Bikash Bhattacharjya
2018 arXiv   pre-print
It is also known that, up to switching isomorphism, there are two signed K_3's, three signed K_4's, and seven signed K_5's.  ...  In this paper, we prove that there are sixteen signed K_6's upto switching ismomorphism.  ...  Each such equivalence class is called a signed graph and is denoted by [G, Σ], where (G, Σ) is any member of the class.  ... 
arXiv:1812.08383v1 fatcat:7qcbqasrindi5orhtskyojzxju

The number of equivalence classes of symmetric sign patterns

Peter J. Cameron, Charles R. Johnson
2006 Discrete Mathematics  
This paper shows that the number of sign patterns of totally non-zero symmetric n-by-n matrices, up to conjugation by permutation and signature matrices and negation, is equal to the number of unlabelled  ...  Let X be the set of all such n-by-n sign patterns, and f 1 (n) the number of equivalence classes under the relation ≡ generated by signature similarity, permutation similarity and negation.  ...  Switching classes and even graphs An even graph is one all of whose valencies are even.  ... 
doi:10.1016/j.disc.2004.10.029 fatcat:c3hefvcdhjblndn6ikz4cupldq

Six signed Petersen graphs, and their automorphisms

Thomas Zaslavsky
2012 Discrete Mathematics  
some of the ideas and methods of signed graph theory.  ...  Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph.  ...  It is a remarkable fact that not just some but every switching equivalence class, and every switching isomorphism class, of signed Petersens has only one minimal signature up to isomorphism.  ... 
doi:10.1016/j.disc.2011.12.010 fatcat:x5wbuewlfzbejkmbno6ifdkoau

Maximum Frustration in Signed Generalized Petersen Graphs [article]

Deepak Sehrawat, Bikash Bhattacharjya
2019 arXiv   pre-print
In this paper, first, we prove that the maximum frustration of generalized Petersen graphs P_n,k is bounded above by n/2 + 1 for (n,k)=1, and this bound is achieved for k=1,2,3.  ...  A signed graph is a simple graph whose edges are labelled with positive or negative signs. A cycle is positive if the product of its edge signs is positive.  ...  We wholeheartedly thank Professor Thomas Zaslavsky for suggesting possible research directions and several comments to improve the presentation of this manuscript.  ... 
arXiv:1905.05548v1 fatcat:ud2o226a5zgflbrec4xbdrcx4u

Homomorphisms of signed graphs: An update [article]

Reza Naserasr, Eric Sopena, Thomas Zaslavsky
2020 arXiv   pre-print
edges of G to (respectively) vertices and edges of H which preserves incidence, adjacency and the signs of closed walks.  ...  Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph (G,σ) to a signed graph (H, π) to be a mapping of vertices and  ...  The work is descendant of an earlier work with Clément Charpentier and has benefited from discussions with him.  ... 
arXiv:1909.05982v2 fatcat:wmqwiyr2kfgubagnv2nntxfaq4

Open problems in the spectral theory of signed graphs

Francesco Belardo, Sebastian M. Cioabă, Jianfeng Wang
2019 The Art of Discrete and Applied Mathematics  
Signed graphs are graphs whose edges get a sign +1 or −1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way.  ...  Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in  ...  Hence, signed graphs from the same switching class share similar graph matrices by means of signature matrices (signature similarity).  ... 
doi:10.26493/2590-9770.1286.d7b fatcat:qofotwluurciznwpptywimvwwy

Spongy Hypercubes [chapter]

Mircea Vasile Diudea
2017 Multi-shell Polyhedral Clusters  
The aim of this paper is to compute the energies of signed spongy hypercubes T□Q and O□Q , where and are tetrahedron and octahedron , respectively .  ...  A spongy hypercube is a Cartesian product of a -connected polyhedral graph and a -dimensional hypercube .  ...  We say that the signed graph is signature switching equivalent (switching equivalent for short) to .  ... 
doi:10.1007/978-3-319-64123-2_11 fatcat:ijydyvjlyjbrtkguzuq2lecnk4

Signed Chromatic Polynomials of Signed Book Graphs [article]

Deepak, Bikash Bhattacharjya
2018 arXiv   pre-print
In this article, we determine the number of different signatures on Book graph up to switching isomorphisms. We also find a recursive formula of the signed chromatic polynomials of signed Book graphs.  ...  In 2015, Matthias Beck and his team developed a computer program in SAGE which efficiently determines the number of signed proper k-colorings for a given signed graph.  ...  Each equivalence class of this equivalence relation is called a signed graph and is denoted by [G, Σ], where (G, Σ) is any member of the class.  ... 
arXiv:1812.08382v1 fatcat:h5oe2ztwivhujicrtuqgbupyom

Perturbations in a signed graph and its index

Zoran StaniĆ
2018 Discussiones Mathematicae Graph Theory  
We also give a partial ordering of signed cacti with common underlying graph by their indices and demonstrate a general method for obtaining lower and upper bounds for the index.  ...  In this paper we consider the behaviour of the largest eigenvalue (also called the index) of signed graphs under small perturbations like adding a vertex, adding an edge or changing the sign of an edge  ...  Let E denote a class of switching equivalent signed graphs and let λ be an eigenvalue belonging to the common spectrum.  ... 
doi:10.7151/dmgt.2035 fatcat:yu2r7ox4brdqjjg4fygk77az24

Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem [article]

Reza Naserasr, Zhouningxin Wang
2021 arXiv   pre-print
For any rational number r=p/q, two notions of circular cliques are presented corresponding to the edge-sign preserving homomorphism and the switching homomorphism.  ...  In this work, we consider the restriction of the circular chromatic number to this class of signed graphs and construct signed bipartite circular cliques with respect to both notions of homomorphisms.  ...  It has also received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 754362.  ... 
arXiv:2109.12618v1 fatcat:xu74q3rmtrhlxl655dtc4crg7q

The chromatic number of a signed graph [article]

Edita Máčajová, André Raspaud, Martin Škoviera
2016 arXiv   pre-print
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph G as a mapping ϕ V(G)→Z such that for any two adjacent vertices u and v the colour ϕ(u) is different from the colour  ...  We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove  ...  of a signed graph" for the main concept of this paper, previously termed "the signed chromatic number".  ... 
arXiv:1412.6349v2 fatcat:pwargcv6enc4joyymbe32uz6jm

Signatures, lifts, and eigenvalues of graphs [article]

Shiping Liu, Norbert Peyerimhoff, Alina Vdovina
2014 arXiv   pre-print
We study the spectra of cyclic signatures of finite graphs and the corresponding cyclic lifts.  ...  Switching equivalence between signatures is an equivalence relation. We denote the corresponding switching class of a signature s by [s].  ...  Note that by Lemma 3, all the signatures in the switching class [s i 0 ] fulfill (5.1).  ... 
arXiv:1412.6841v1 fatcat:r4ttbitf7jhqvisunv33jntgaa

Concepts of signed graph coloring [article]

Eckhard Steffen, Alexander Vogel
2020 arXiv   pre-print
This paper surveys recent development of concepts related to coloring of signed graphs. Various approaches are presented and discussed.  ...  One of them pointed out that there is a short survey on some aspects of coloring signed graphs by Lynn Takeshita [45] .  ...  We will follow the definitions stated in [37] and define homomorphisms on signed graphs as homomorphisms on switching classes of signed graphs.  ... 
arXiv:1909.09381v2 fatcat:uveiuweurrblxavntkdulqhb2m
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