A theta graph is a homeomorph of K 2;3. In an embedded planar graph the local rotation at one degree-three vertex of a theta graph determines the local rotation at the other degree-three vertex. Using this observation, we give a characterization of planar graphs in terms of balance in an associated signed graph whose vertices are K 1;3 sub-graphs and whose edges correspond to theta graphs.

A signed graph (G,Σ) is an undirected graph G together with an assignment of signs (positive or negative) to all its edges, where Σ denotes the set of negative edges. Two signatures are said to be equivalent if one can be obtained from the other by a sequence of resignings (i.e. switching the sign of all edges incident to a given vertex). Extending the notion of usual graph homomorphisms, homom...

We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a nonsplit link for every signed embedding. With one ...

We prove inequalities between the densities of various bipartite subgraphs in signed graphs. One of the main inequalities is that the density of any bipartite graph with girth 2r cannot exceed the density of the 2r-cycle. This study is motivated by the Simonovits–Sidorenko conjecture, which states that the density of a bipartite graph F with m edges in any graph G is at least the m-th power of ...

In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.

For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. Continuing the study of these matrices associated to an oriented hypergraph, several related structures are investigated including: the incidence dual, the intersection graph (line graph), and the 2-section. The intersection graph is show...

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13–28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which are known to occur in two types – either balanced circuits or pairs of disjoint unbalanced circuits connecte...

A graph G is signed if each edge is assigned + or −. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign − if and only if its endpoints are in different parts. The Edwards-Erdös bound states that every graph with n vertices and m edges has a balanced subgraph with at least m 2 +n−1 4 edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max C...

The normalized Laplacian of a graph was introduced by F.R.K. Chung and has been studied extensively over the last decade. In this paper, we introduce the notion of the normalized Laplacian of signed graphs and extend some fundamental concepts of the normalized Laplacian from graphs to signed graphs.

An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families – balanced, balanceable, and unbalanceable – and we obtain a complete classificati...