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Characterisations of interval graphs, comparability graphs, co-comparability graphs, permutation graphs, and split graphs in terms of linear orderings of the vertex set are presented. As an application, it is proved that interval graphs,…
As in algebraic geometry, an effective divisor class on a vertex-weighted graph is called special if also its residual class is effective. We study the question, when this is true already on the level of divisors; that is, when there exists…
We attach to each $\langle 0, \vee \rangle$-semilattice a graph $\boldsymbol{G}_{\boldsymbol{S}}$ whose vertices are join-irreducible elements of $\boldsymbol{S}$ and whose edges correspond to the reflexive dependency relation. We study…
In this paper, we study polynomial-like elements in vector spaces equipped with group actions. We first define these elements via iterated difference operators. In the case of a full rank lattice acting on an Euclidean space, these…
Let $R$ be a ring with unity. The non-zero divisor graph of $R$, $\Phi(R)$, is the graph with vertex set $R\backslash \{0,1,-1\}$, and two vertices $x$ and $y$ are adjacent if and only if either $xy$ or $yx$ is non-zero. In this article we…
The $\lambda$-core vertices of a graph correspond to the non-zero entries of some eigenvector of $\lambda$ for a universal adjacency matrix $\mathbf{U}$ of the graph. We define a partition of the vertex set $V$ based on the $\lambda$-core…
Let $R$ be a finite product of integral domains and $D$ be a union of prime ideals (it is possible that $R$ is just an integral domain). Let $n \geq 1$ be a positive integer. This paper introduces the $n$-total graph of a $(R, D)$. The…
In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We…
We look for a non-zero $(0, 1)$-vector in the row space of the adjacency matrix $A(\Gamma)$ of a graph $\Gamma,$ provided $\Gamma$ has at least one edge. Akbari, Cameron, and Khosrovshahi conjectured that there exists a non-zero…
We study the nullspace of the adjacency matrix of split graphs, whose vertex set can be partitioned into a clique and an independent set. We introduce the clique-kernel, a subspace that decides whether clique vertices lie in the support of…
In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of…
We determine the optimum topology of quasi-one dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs endowed with a metric and a multiparticle Hamiltonian acting on the edges,…
Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with…
The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in…
Block designs are combinatorial structures in which each pair of a set of varieties appears together in a fixed number of blocks. Complete graphs are graphs in which every pair of vertices are adjacent. We present some new constructions of…
We show how arbitrary unit cells of periodic materials can be represented as graphs whose nodes represent atoms and whose weighted edges represent tunneling connections between atoms. Further, we present methods to calculate the band…
We consider general integrable systems on graphs as discrete flat connections with the values in loop groups. We argue that a certain class of graphs is of a special importance in this respect, namely quad-graphs, the cellular…
Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not…
We introduce the factorization graph of a finite group and study its connectedness and forbidden structures. We characterize all finite groups with connected factorization graphs and classify those with connected bipartite factorization…
The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected…