Related papers: Row graphs of Toeplitz matrices

We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number…

Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by…

Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a 2-connected,…

The subclass of Euler graphs with only one type of cycles under (mod 4) operation was studied in Part-1 of this series. It was established that such graphs under regularity are nonexistent for degree >2. Here we consider the subclass of…

We continue studying Thomassen's conjecture (every 4-connected line graph has a Hamilton cycle) in the direction of a recently shown equivalence with Jackson's conjecture (every 2-connected claw-free graph has a Tutte cycle), and we extend…

For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and…

We study a question that lies at the intersection of classical research subjects in Topological Graph Theory and Graph Drawing: Computing a drawing of a graph with a prescribed number of crossings on a given set $S$ of points, while…

Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible…

A hole in a graph is an induced subgraph which is a cycle of length at least four. A graph is chordal if it contains no holes. Following McKee and Scheinerman (1993), we define the chordality of a graph $G$ to be the minimum number of…

We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most…

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…

An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} \approx 1.5849^n$ maximal induced matchings, and this bound is best possible. We…

Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the…

Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define the \emph{total $k$-cut complex} of a graph $G$ to be the simplicial complex whose facets are the complements of independent sets of size $k$ in $G$. We study the…

Given a connected graph $G\ $of order $n$ and a nonnegative symmetric matrix $A=\left[ a_{i,j}\right] $ of order $n,$ define the function $F_{A}\left( G\right) $ as% \[ F_{A}\left( G\right) =\sum_{1\leq i<j\leq n}d_{G}\left( i,j\right)…

Minimal prime graphs (MPGs) are a special class of prime graphs (also known as Gruenberg-Kegel graphs) associated with finite solvable groups. A graph is an MPG if it has at least two vertices, is connected, its complement is triangle-free…

Connection matrices were introduced by Freedman, Lovasz and Schrijver [1], who used them to characterize graph homomorphism functions. The goal of this note is to determine the exact rank of these matrices. The result can be rephrased in…

A graph $G$ of constant link $L$ is a graph in which the neighborhood of any vertex induces a graph isomorphic to $L$. Given two different graphs, $H$ and $G$, the induced Tur\'an number ${\rm ex}(n; H, G{\rm -ind})$ is defined as the…

A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges…

In this brief communication, we investigate the cospectral as well integral chain graphs for Seidel matrix, a key component to study the structural properties of equiangular lines in space. We derive a formula that allows to generate an…