Related papers: Cordial Digraphs
We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs,…
In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament…
Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,\sigma)$ is a weighted…
We introduce two new measures of the noncordiality of a graph. We then calculate the values of these measures for various families of noncordial graphs. We also determine exactly which of the M\"obius ladders are cordial.
The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U,…
We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We…
A graph of order $n$ is distance magic if it admits a bijective labeling of its vertices with integers from $1$ to $n$ such that each vertex has the same sum of the labels of its neighbors. This paper contributes to the long term project of…
A subgraph $G'$ of a graph $G$ is nice if $G-V(G')$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex $u$ of a cubic graph is nice if $u$ and…
A dissociation set of a graph is a set of vertices which induces a subgraph with maximum degree less than or equal to one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we study the…
Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is…
An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield…
Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in…
Homophily is a graph property describing the tendency of edges to connect similar nodes; the opposite is called heterophily. It is often believed that heterophilous graphs are challenging for standard message-passing graph neural networks…
A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it admits an orientation such that the out-neighborhood of every vertex is a clique in G. The class of 1-p.o. graphs forms a common generalization of the classes of…
Homophily is a graph property describing the tendency of edges to connect similar nodes. There are several measures used for assessing homophily but all are known to have certain drawbacks: in particular, they cannot be reliably used for…
A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints,…
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said…
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc…
Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…
Given a graph G=(V, E), a vertex is said to ve-dominate an edge if it is either incident with the edge or adjacent to one of its endpoints. A set of vertices is a ve-dominating set if it ve-dominates every edge of the graph. We introduce…