Related papers: Negative closed walks in signed graphs: A note
A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of…
A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A…
A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen…
Let $G_S$ be a graph with loops attached at each vertex in $S \subseteq V(G).$ In this article, we develop exact formulae for the number of closed $3$- and $4$-walks on $G_S$ in terms of vertex degrees and certain elementary subgraphs of…
A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed…
In a signed graph each edge has a sign, $+1$ or $-1$. We introduce in the present paper a new definition of connection in a signed graph by the existence of both positive and negative chains between vertices. We prove some results and…
A signed graph $(G, \Sigma)$ is a graph $G$ and a subset $\Sigma$ of its edges which corresponds to an assignment of signs to the edges: edges in $\Sigma$ are negative while edges not in $\Sigma$ are positive. A closed walk of a signed…
In this note we prove that every closed graph $G$ is up to isomorphism a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.
A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles.…
Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based…
Signed graphs have their edges labeled either as positive or negative. Here we introduce two types of signed distance matrix for signed graphs. We characterize balance in signed graphs using these matrices and we obtain explicit formulae…
A \emph{signed graph} $(G, \sigma)$ is a graph $G$ together with an assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between…
In this paper we introduce the nullity of signed graphs, and give some results on the nullity of signed graphs with pendant trees. We characterize the unicyclic signed graphs of order n with nullity n-2; n-3; n-4; n-5 respectively.
Let $G^{\sigma}=(G,\sigma)$ be a signed graph and $A(G,\sigma)$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $\eta(G,\sigma)$ be the nullity of $(G,\sigma)$. He et al. [Bounds for the matching number and…
Graph-walking automata (GWA) are a model for graph traversal using finite-state control: these automata move between the nodes of an input graph, following its edges. This paper investigates the effect of node-replacement graph…
The $k$-th exact-distance graph, of a graph $G$ has $V(G)$ as its vertex set, and $xy$ as an edge if and only if the distance between $x$ and $y$ is (exactly) $k$ in $G$. We consider two possible extensions of this notion for signed graphs.…
A signed graph $\Sigma = (G, \sigma)$ is a graph where the function $\sigma$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $\Sigma$, denoted by $A(\Sigma)$, is defined canonically. In a recent…
We work with combinatorial maps to represent graph embeddings into surfaces up to isotopy. The surface in which the graph is embedded is left implicit in this approach. The constructions herein are proof-relevant and stated with a subset of…
A signed graph is a graph in which each edge has a positive or negative sign. In this article, first we characterize the distance compatibility in the case of a connected signed graph and discussed the distance compatibility criterion for…
This paper explains the periodicity of the Grover walk on finite graphs. We characterize the graphs to induce 2, 3, 4, 5-periodic Grover walk and obtain a necessary condition of the graphs to induce an odd-periodic Grover walk.