Related papers: On the arithmetic-geometric index of graphs

In 1975 Erd\H{o}s initiated the study of the following very natural question. What can be said about the chromatic number of unit distance graphs in $\mathbb{R}^2$ that have large girth? Over the years this question and its natural…

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$.…

Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in…

The {\em acyclic chromatic number} of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The {\em acyclic chromatic index} is the analogous graph parameter for edge…

To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference…

Given a connected graph $G(V, E)$, the edge dimension, denoted $\mathrm{edim}(G)$, is the least size of a set $S \subseteq V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise distinct tuples of…

The chromatic edge-stability number ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph $G'$ with $\chi(G')=\chi(G)-1$. Edge-stability critical graphs are introduced as the graphs…

This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…

Chemical graphs are simple undirected connected graphs, where vertices represent atoms in a molecule and edges represent chemical bonds. A degree-based topological index is a molecular descriptor used to study specific physicochemical…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{deg(v) + deg(u)}$ of all edges $uv$ of $E (G)$, where $deg (v)$ denotes the degree of a vertex $v$ in $V (G)$. In this note we generalize results of [L. Zhong, The…

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in…

The edge domination number $\gamma_e(G)$ of a graph $G$ is the minimum size of a maximal matching in $G$. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been…

This work introduces the concept of \emph{upper-critical graphs}, in a complementary way of the conventional (lower)critical graphs: an element $x$ of a graph $G$ is called \emph{critical} if $\chi(G-x)<\chi(G)$. It is said that $G$ is a…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. Let $G$ be a connected graph…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which…

The geometric--arithmetic indices are widely considered in the chemical graph theory in the last decade. The reason of introducing new indices is to gain prediction of target properties of considered molecules that is better than the…

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is…

In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…