Related papers: Characterization of Line-Consistent Signed Graphs
The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every maximal clique of H. A graph is claw-free if no vertex has three pairwise non-adjacent neighbors. The characterization of claw-free graphs that are…
Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph…
The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts.…
A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs…
Chordal graphs and chordal bigraphs enjoy beautiful characterizations, in terms of forbidden subgraphs, vertex/edge orderings, vertex/edge separating sets, and tree-like representations. In this paper, we introduce chordal signed graphs and…
A signed graph $(G,\Sigma)$ is a graph $G$ together with a set $\Sigma \subseteq E(G)$ of negative edges. A circuit is positive if the product of the signs of its edges is positive. A signed graph $(G,\Sigma)$ is balanced if all its…
One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…
A signed graph is a graph whose edges are labeled either as positive or negative. The concept of vector valued switching and balancing dimension of signed graphs were introduced by S. Hameed et al. In this paper, we deal with the balancing…
Structural balance theory assumes triads in networks to gravitate towards stable configurations. The theory has been verified for undirected graphs. Since real-world networks are often directed, we introduce a novel method for considering…
We consider signed networks in which connections or edges can be either positive (friendship, trust, alliance) or negative (dislike, distrust, conflict). Early literature in graph theory theorized that such networks should display…
A signed graph has edge signs. A gain graph has oriented edge gains drawn from a group. We define the combination of the two for the abelian case, in which each oriented edge of a signed graph has a gain from an abelian group, concentrating…
A signed graph is a graph with a function that assigns a label of positive or negative to each edge. The sign of a circle is the product of the signs of its edges; a graph is balanced if all of its circles are positive. A set of edges whose…
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…
Let $G$ be a graph and let $A(G)$ be the adjacency matrix of $G$. The signature $s(G)$ of $G$ is the difference between the positive inertia index and the negative inertia index of $A(G)$. Ma et al. [Positive and negative inertia index of a…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
We prove that a connected graph contains a circuit---a closed walk that repeats no edges---through any $k$ prescribed edges if and only if it contains no odd cut of size at most $k$.
A simple $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$ such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex…
The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed…
A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another…