Related papers: Resolving Open Problems on the Euler Sombor Index

The online dominating set problem is an online variant of the minimum dominating set problem, which is one of the most important NP-hard problems on graphs. This problem is defined as follows: Given an undirected graph $G = (V, E)$, in…

Performing inference in graphs is a common task within several machine learning problems, e.g., image segmentation, community detection, among others. For a given undirected connected graph, we tackle the statistical problem of exactly…

The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set $Q\subseteq V$…

The $\sigma$-irregularity index of a graph is defined as the sum of squared degree differences over all edges and provides a sensitive measure of structural heterogeneity. In this paper, we study the problem of maximizing $\sigma(T)$ among…

A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…

Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is $BID(G)=\sum_{uv\in E(G)}$ $\Psi(d_{u},d_{v})$, where $uv$ is the edge connecting the vertices $u,v$ of the graph…

In this paper, we presented a study of topological indices on trees, where we show a relationship with irregularity of Albertson index and minimum, maximum degrees $\delta,\Delta$ of graph $G$, where contribute vital roles in determining…

In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity…

Consider an undirected graph $G = (VG, EG)$ and a set of six \emph{terminals} $T = \set{s_1, s_2, s_3, t_1, t_2, t_3} \subseteq VG$. The goal is to find a collection $\calP$ of three edge-disjoint paths $P_1$, $P_2$, and $P_3$, where $P_i$…

The vertex isoperimetric number of a graph $G=(V,E)$ is the minimum of the ratio $|\partial_{V}U|/|U|$ where $U$ ranges over all nonempty subsets of $V$ with $|U|/|V|\le u$ and $\partial_{V}U$ is the set of all vertices adjacent to $U$ but…

The degree of a based graph is the number of essential nonbasepoint vertices after generic perturbation. Hatcher--Vogtmann's degree theorem states that the subcomplex of Auter space of graphs of degree at most d is (d-1)-connected. We…

Let $G$ be a connected graph. The revised edge Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is…

The eccentric sequence of a connected graph $G$ is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. The unique trees that…

A set cover of a hypergraph $H$ is a set of vertices intersecting every hyperedge. In the minimum sum set cover problem, vertices are selected one by one; each edge pays the position of the first vertex that hits it, and the objective is to…

We show that a problem of deleting a minimum number of vertices from a graph to obtain a graph embeddable on a surface of a given Euler genus is solvable in time $2^{C_g \cdot k^2 \log k} n^{O(1)}$, where $k$ is the size of the deletion…

We study the minimal homogeneous generating sets of the Eulerian ideal associated with a simple graph and its maximal generating degree. We show that the Eulerian ideal is a lattice ideal and use this to give a characterization of binomials…

Dynamic Connectivity is a fundamental algorithmic graph problem, motivated by a wide range of applications to social and communication networks and used as a building block in various other algorithms, such as the bi-connectivity and the…

This paper focuses on establishing the existence of a class of steady solutions, termed least total curvature solutions, to the incompressible Euler system in a strip. The solutions obtained in this paper complement the least total…

We introduce the \emph{ID-index} of a finite simple connected graph. For a graph $G=(V,\ E)$ with diameter $d$, we let $f:V\longrightarrow \mathbb{R}$ assign \emph{ranks} to the vertices, then under $f$, each vertex $v$ gets a…

The Euler tour technique is a classical tool for designing parallel graph algorithms, originally proposed for the PRAM model. We ask whether it can be adapted to run efficiently on GPU. We focus on two established applications of the…