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Signatures and signed switching classes
1986
Journal of combinatorial theory. Series B (Print)
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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]
2010
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2006
Discrete Mathematics
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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
2012
Discrete Mathematics
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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]
2019
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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
2019
The Art of Discrete and Applied Mathematics
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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]
2017
Multi-shell Polyhedral Clusters
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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]
2018
arXiv
pre-print
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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
2018
Discussiones Mathematicae Graph Theory
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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]
2021
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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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