Related papers: Solution to an open problem on the closeness of gr…
A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…
A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we consider several problems of graph theory and developed optimal algorithms to solve such problems on cactus graphs. The running time of…
In the Telephone Broadcasting problem, the goal is to disseminate a message from a given source vertex of an input graph to all other vertices in the minimum number of rounds, where at each round, an informed vertex can send the message to…
A cactus is a connected graph in which any two cycles have at most one common vertex. We determine the unique graph that maximizes the distance spectral radius over all cacti with fixed numbers of vertices and cycles, and thus prove a…
Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an…
The subpath number of a graph G is defined as the total number of subpaths in G, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper [5], where we…
Networks are inherently vulnerable to vertex failures, making the analysis of their structural robustness a fundamental problem in graph theory. In this study, we investigate the closeness and vertex residual closeness of graphs, with a…
We study the problem of exploring all vertices of an undirected weighted graph that is initially unknown to the searcher. An edge of the graph is only revealed when the searcher visits one of its endpoints. Beginning at some start node, the…
Consider the continuum of points on the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network…
A graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. Denote by $Cact(n;t)$ the set of connected cacti possessing $n$ vertices and $t$ cycles. In this paper, we show that there are some errors in [J. Du, G. Su,…
Closeness is one of the most studied characteristics of networks. Residual closeness is a very sensitive measure of graphs robustness. Additional closeness is a measure of growth potentials of networks. In this article we calculate the…
The edge-Wiener index $W_e(G)$ of a connected graph $G$ is the sum of distances between all pairs of edges of $G$. A connected graph $G$ is said to be a cactus if each of its blocks is either a cycle or an edge. Let $\mathcal{G}_{n,t}$…
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…
In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric…
Dinits-Karzanov-Lomonosov showed that it is possible to encode all minimal edge cuts of a graph by a tree-like structure called a cactus. We show here that minimal edge cuts separating ends of the graph rather than vertices can be `encoded'…
In this article we consider networks, which for a given time period can have one link broken. Which new link should we build so the closeness of the resulting network satisfies some optimal criteria? We consider different criteria for…
The problem of realizing finite metric spaces in terms of weighted graphs has many applications. For example, the mathematical and computational properties of metrics that can be realized by trees have been well-studied and such research…
Let $\prod(G)$ be Multiplicative Zagreb index of a graph G. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which has been the interest of researchers in the filed of material…
The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$…
We study the Telephone Broadcasting problem in graphs with restricted structure. Given a designated source in an undirected graph, the goal is to disseminate a message to all vertices in the minimum number of rounds, where in each round…