Related papers: On singular signed graphs with nullspace spanned b…
An edge uv in a graph \Gamma\ is directionally 2-signed (or, (2,d)-signed) by an ordered pair (a,b), a,b in {+,-}, if the label l(uv) = (a,b) from u to v, and l(vu) = (b,a) from v to u. Directionally 2-signed graphs are equivalent to…
Let \Gamma be a signed graph and let A(\Gamma) be the adjacency matrix of \Gamma. The nullity of \Gamma is the multiplicity of eigenvalue zero in the spectrum of A(\Gamma). In this paper we characterize the signed graphs of order n with…
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-Erd\"os bound…
A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges…
A circulant graph is a simple graph whose adjacency matrix can be represented in the form of a circulant matrix, while a nut graph is considered to be a graph whose null space is spanned by a single full vector. In a previous study by…
We introduce new sufficient conditions for intrinsic knotting and linking. A graph on n vertices with at least 4n-9 edges is intrinsically linked. A graph on n vertices with at least 5n-14 edges is intrinsically knotted. We also classify…
For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…
A nut graph is a non-trivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. It was recently shown by the authors that there exists a $d$-regular circulant nut graph of order $n$ if and…
Signed networks have been a topic of recent interest in the network control community as they allow studying antagonistic interactions in multi-agent systems. Although dynamical characteristics of signed networks have been well-studied,…
The niche graph of a digraph $D$ is the (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if $N^+_D(x) \cap N^+_D(y) \neq \emptyset$ or $N^-_D(x) \cap…
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…
We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two…
Two competing types of interactions often play an important part in shaping system behavior, such as activatory or inhibitory functions in biological systems. Hence, signed networks, where each connection can be either positive or negative,…
Signed graphs are widely used to analyze complex systems such as social, political, and biological networks. The notion of balance, a key concept of signed graphs, reflects the stability of relationships. While it has been extensively…
Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned of hydrogen atoms, are chemical graphs. A chemical graph is a connected simple graph of maximum degree $3$ or less. A nut graph is a connected simple graph with a…
Signed graphs have been introduced to enrich graph structures expressing relationships between persons or general social entities, introducing edge signs to reflect the nature of the relationship, e.g., friendship or enmity. Independently,…
A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms…
A signed graph is a graph $G$ associated with a mapping $\sigma: E(G)\to \{-1,+1\}$, denoted by $(G,\sigma)$. A $cycle$ of $(G,\sigma)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges,…
A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four…
We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of…