Related papers: New measures of graph irregularity
Let G be an n-vertex graph with m edges. The degree deviation measure of G is defined as s(G)=sum v in V(G)|degG(v)-(2m/n)|, where n and m are the number of vertices and edges of G, respectively. The aim of this paper is to prove the…
The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large…
Trophic coherence and non-normality are both ways of describing the overall directionality of directed graphs, or networks. Trophic coherence can be regarded as a measure of how neatly a graph can be divided into distinct layers, whereas…
Measurement incompatibility--the impossibility of jointly measuring certain quantum observables--is a fundamental resource for quantum information processing. We develop a graph-theoretic framework for quantifying this resource for large…
The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the…
This paper introduces a simple measure of a concordance pattern among observed outcomes along a network, i.e., the pattern in which adjacent outcomes tend to be more strongly correlated than non-adjacent outcomes. The graph concordance…
Understanding causal relationships among the variables of a system is paramount to explain and control its behavior. For many real-world systems, however, the true causal graph is not readily available and one must resort to predictions…
A typical result in graph theory says that a graph $G$, satisfying certain conditions, has some property $\cal P$. Once such a theorem is established, it is natural to ask how strongly $G$ satisfies $\cal P$. Can one strengthen the result…
In this paper we investigate a parameter of graphs, called the circular altitude, introduced by Peter Cameron. We show that the circular altitude provides a lower bound on the circular chromatic number, and hence on the chromatic number, of…
A new measure $c(e)$ of the centrality of an edge $e$ in an undirected graph $G$ is introduced. It is based on the variation of the Kemeny constant of the graph after removing the edge $e$. The new measure is designed in such a way that the…
The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the…
We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone,…
In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.
Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove…
We present a unified approach to compute the total vertex irregularity strength (tvs) of various graphs, employing a novel technique recently proposed by Barra et al. For graphs such as cycles, paths, prisms, wheels, complete graphs, helm…
The total $\sigma$-irregularity is given by $ \sigma_t(G) = \sum_{\{u,v\} \subseteq V(G)} \left(d_G(u) - d_G(v)\right)^2, $ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing…
The variation of the Randi\'c index $ R'(G) $ of a graph $G$ is defined by\ $R(G) = \sum_{uv \in E(G)}\frac 1{\max \{d(u) d(v)\}}$, where $d(u)$ is the degree of vertex $u$ and the summation extends over all edges $uv$ of $G$. Let $G(k,n)$…
In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of…
In this paper, we define several measures induced by a finite directed graph. The study themselves is interesting ont only in the noncommutative probability point of view but also in the algebraic structure point of view, since to define…
We define the $d$-defective incidence chromatic number of a graph, generalizing the notion of incidence chromatic number, and determine it for some classes of graphs including trees, complete bipartite graphs, complete graphs, and…