Related papers: The Antisymmetric Line Graph

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…

Statistical inference on graphs is a burgeoning field in the applied and theoretical statistics communities, as well as throughout the wider world of science, engineering, business, etc. In many applications, we are faced with the reality…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$.…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right] $, write $A_{\alpha}\left( G\right) $ for the matrix \[ A_{\alpha}\left( G\right)…

For a graphical property $\mathcal{P}$ and a graph $G$, a subset $S$ of vertices of $G$ is a $\mathcal{P}$-set if the subgraph induced by $S$ has the property $\mathcal{P}$. The domination number with respect to the property $\mathcal{P}$,…

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes…

The distinguishing number of a permutation group $G\leqslant\Sym(\Omega)$ is the minimum number of colours needed to colour $\Omega$ in such a way that the only colour preserving element of $G$ is the identity. The distinguishing number of…

The nodal edge count of an eigenvector of the Laplacian of a graph is the number of edges on which it changes sign. This quantity extends to any real symmetric $n\times n$ matrix supported on a graph $G$ with $n$ vertices. The average nodal…

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors.…

A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is…

In this work, we delve into the study of the 2-switch-degree of a graph $G$, which is nothing more than the degree of $G$ as a vertex of the realization graph $\mathcal{G}(d)$ associated with the degree sequence $d$ of $G$. We explore the…

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular graph for some…

For a graph $G$, the $\gamma$-graph of $G$, $G(\gamma)$, is the graph whose vertices correspond to the minimum dominating sets of $G$, and where two vertices of $G(\gamma)$ are adjacent if and only if their corresponding dominating sets in…

Let $G$ be a $k$-degenerate graph of order $n.$ It is well-known that $G\ $has no more edges than $S_{n,k},$ the join of a complete graph of order $k$ and an independent set of order $n-k.$ In this note it is shown that $S_{n,k}$ is…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_\mu(G):=A(G)-\mu D(G)$. Two graphs $G_1$ and $G_2$ are called \emph{degree-similar} if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M…

A set of edges $\Gamma$ of a graph $G$ is an edge dominating set if every edge of $G$ intersects at least one edge of $\Gamma$, and the edge domination number $\gamma_e(G)$ is the smallest size of an edge dominating set. Expanding on work…

Let $G$ and $H$ be two simple graphs. A bijection $\phi:V(G)\rightarrow V(H)$ is called an isomorphism between $G$ and $H$ if $(\phi v_i)(\phi v_j)\in E(H)$ $\Leftrightarrow$ $v_i v_j\in E(G)$, $\forall v_i,v_j \in V(G)$. In the case that…

In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $\Sigma = (G,\sigma)$ is a graph $G$ together…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

The graph isomorphism is to determine whether two graphs are isomorphic. A closely related problem is automorphism detection, where an isomorphism between two graphs is a bijection between their vertex sets that preserves adjacency, and an…