There are eight ways that signs can be assigned to the sides of a triangle. An odd number of negative signs makes an unbalanced triangle, according to Fritz Heider's theory.
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.
A signed graph is balanced if the product of edge signs around every cycle is positive. The name "signed graph" and the notion of balance appeared first in a mathematical paper of Frank Harary in 1953.[1] Dénes Kőnig had already studied equivalent notions in 1936 under a different terminology but without recognizing the relevance of the sign group.[2]
At the Center for Group Dynamics at the University of Michigan, Dorwin Cartwright and Harary generalized Fritz Heider's psychological theory of balance in triangles of sentiments to a psychological theory of balance in signed graphs.[3][4]
Signed graphs have been rediscovered many times because they come up naturally in many unrelated areas.[5] For instance, they enable one to describe and analyze the geometry of subsets of the classical root systems. They appear in topological graph theory and group theory. They are a natural context for questions about odd and even cycles in graphs. They appear in computing the ground state energy in the non-ferromagnetic Ising model; for this one needs to find a largest balanced edge set in Σ. They have been applied to data classification in correlation clustering.
^Harary, Frank (1955), "On the notion of balance of a signed graph", Michigan Mathematical Journal, 2: 143–146, MR 0067468, archived from the original on 2013-04-15
^Kőnig, Dénes (1936), Akademische Verlagsgesellschaft (ed.), Theorie der endlichen und unendlichen Graphen
^Cartwright, D.; Harary, Frank (1956). "Structural balance: a generalization of Heider's theory" (PDF). Psychological Review. 63 (5): 277–293. doi:10.1037/h0046049. PMID 13359597.
^Steven Strogatz (2010), The enemy of my enemy, The New York Times, February 14, 2010
^Zaslavsky, Thomas (1998), "A mathematical bibliography of signed and gain graphs and allied areas", Electronic Journal of Combinatorics, 5, Dynamic Surveys 8, 124 pp., MR 1744869.
In the area of graph theory in mathematics, a signedgraph is a graph in which each edge has a positive or negative sign. A signedgraph is balanced if...
3-cycles in a signedgraph. The sign of a path in a graph is the product of the signs of its edges. They considered cycles in a signedgraph representing...
signedgraph is a generalization of the oriented incidence matrix. It is the incidence matrix of any bidirected graph that orients the given signed graph...
graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject...
stochastic block model is a generative model for random graphs. This model tends to produce graphs containing communities, subsets of nodes characterized...
skew-symmetric graph is the double covering graph of a bidirected graph. A bidirected graph may be regarded as an orientation of a signedgraph, similarly...
developers’ applications to interact with the Graph API on behalf of Facebook users, and it provides a single-sign on mechanism across web, mobile, and desktop...
the gain of e (in some indicated direction). A gain graph is a generalization of a signedgraph, where the gain group G has only two elements. See Zaslavsky...
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian...
mathematical sociology a signedgraph may be used to represent a social network that may or may not be balanced, depending upon the signs found along cycles...
Theorem for signed graphs, which was published by Frank Harary in 1953. A signedgraph is called balanced if the product of the signs of all relations in...
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges...
the HCS clustering algorithm. Signedgraph models: Every path in a signedgraph has a sign from the product of the signs on the edges. Under the assumptions...
of the combinatorial essentials of a gain graph and in particular of a signedgraph. Formally, a biased graph Ω is a pair (G, B) where B is a linear class...
In graph theory, the hypercube graph Qn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q3...
matroid of G . {\displaystyle G~.} A signedgraph, whose edges are labeled by signs, and a gain graph, which is a graph whose edges are labeled orientably...
connected signedgraph G. Bound (a) was improved for special classes of graphs: triangle-free graphs, graphs of given maximum degree, H-free graphs, etc....
equation Parallel curve (also known as offset curve) Signed arc length Signed area Signed measure Signed volume Chan, T.; Zhu, W. (2005). Level set based...
American mathematician, who specialized in graph theory. He was widely recognized as one of the "fathers" of modern graph theory. Harary was a master of clear...
A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that...
spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are...
In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some...
a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes...
topological graph theory, a ribbon graph is a way to represent graph embeddings, equivalent in power to signed rotation systems or graph-encoded maps...
Global Information