Related papers: Adjacent cycle-chains are $e$-positive

A chorded cycle in a graph $G$ is a cycle on which two nonadjacent vertices are adjacent in the graph $G$. In 2010, Gao and Qiao independently proved a graph of order at least $4s$, in which the neighborhood union of any two nonadjacent…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges…

For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly…

\noindent In this paper, we show that for any positive integers $r$, $k$, $\Theta$, and $\Gamma$ such that $k \geq 2$ and $r \geq k + \Gamma$, there exists a connected graph $G$ for which $$\begin{array}{llcr} \omega (G) = \chi (G) = k, &…

We provide a matrix-based formula for the Tutte symmetric function of a graph. In particular, for any graph $G$ with a designated head and tail vertex, we describe an infinite matrix $M_G$ from which the Tutte symmetric function can be…

For a given function from a set to itself, we can define a directed graph called the functional graph, where the vertices are the elements of the set, and the edges are all the pairs of inputs and outputs for the function. In this article…

In this paper we prove a conjecture by Wocjan, Elphick and Anekstein (2018) which upper bounds the sum of the squares of the positive (or negative) eigenvalues of the adjacency matrix of a graph by an expression that behaves monotonically…

The {\em disjointness graph} of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph $G$ of any system of segments in the plane is {\em…

Given two graphs G and H its 1-{\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition…

In a balanced graph decomposition, every vertex of the host graph appears in the same number of blocks. We propose the use of colored loops as a framework for unifying various other types of local balance conditions in graph decompositions.…

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed…

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a…

Closeness is an important measure of network centrality. In this article we will calculate the closeness of graphs, created by using operations on graphs. We will prove a formula for the closeness of shadow graphs. We will calculate the…

We prove that for any tree with a vertex of degree at least six, its chromatic symmetric function is not $e$-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes…

In this note we prove that the constant and equivariant cyclic cohomology of algebras coincide. This shows that constant cyclic cohomology is rich and computable.

A graph is said to be orthogonalisable if the set of real symmetric matrices whose off-diagonal pattern is prescribed by its edges contains an orthogonal matrix. We determine some necessary and some sufficient conditions on the sizes of the…

Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $\chi_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the…

We obtain the Schur positivity of spider graphs of the forms $S(a,2,1)$ and $S(a,4,1)$, which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find…

A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of…