# Finding the mother vertex in a graph

Networks or graphs are pivotal in so many real-world applications such as fraud prevention systems. search engines, recommendation systems, social networks and a lot more. The search for the mother vertex of a graph aids in understanding the accessibility of a given vertex or collection of vertices.

What is the meaning of the mother vertex in a given graph?

In a given graph G = (V, E), a mother vertex v has a pathway for all other vertices in the specified graph to enable the connection. All other vertices in the graph can be accessed via the mother vertex. A mother vertex is quite common in directed graphs but is also applicable in undirected networks. We will briefly explore mother vertices in different network examples

Directed graph: Directed graphs or DiGraphs do hold directed edges or nodes. These edges can be unidirectional or bidirectional. For DiGraphs, self-loops are permissible but parallel edges are not. Mother vertex exists in directed graphs and there can be multiple of these as shown below.  Based on the directed graph below, nodes  and  are the mother vertex.

# Exploring Dijkstra’s shortest path algorithm

Several graph algorithms can help reveal hidden patterns in connected data. These algorithms can be classified into several categories such as approximations (e.g clustering), assortativity (e,g average neighbour degree), communities (e.g K-Clique) and centrality (e.g shortest path). In this blog, we will be looking at one of the most popular shortest path algorithms known as the Dijkstra’s algorithm. Exploring an example table and code implementation for this algorithm. Shortest path algorithm can be relevant in a traffic network situation a user desires to discover the fastest way to move from a source to a destination. It is an iterative algorithm that provides us with the shortest path from an origin node to all other nodes in the graph. This algorithm can work in weighted and unweighted graph scenarios.

# Different ways of representing Graphs

Graph data can be represented in different formats for onward computation. The choice of the graph representation hugely relies on the density of the graph, space required, speed and weight of edges. The main ways a graph can be represented are as an adjacency matrix, incidence matrix, adjacency list and incidence list.

Adjacency Matrix: This is one of the most popular ways a graph is represented. One of the core aims of this matrix is to assess if the pairs of vertices in a given graph are adjacent or not. In an adjacency matrix, row and columns represent vertices. The row sum equals the degree of the vertex that the row represents. It is advisable to use the adjacency matrix for weighted edges.  You replace the standard ‘1’ with the respective weight.  It is easier to represent directed graphs with edge weights through an adjacency matrix. Adjacency matrix works best with dense graphs. As dense graphs usually have twice the size of the edges to the given nodes.

# Introduction to a graph data structure in Python

Graphs play an important part in intelligent technologies such as recommendation engines, search engines, fraud detection applications, network mapping, customer journey applications, latency evaluation and dependency management. Graphs data structures are also powerful in social networks such as Twitter, Facebook, LinkedIn and a few others. Graphs in the context of the social graph are used to recommend friends, events, company pages, jobs and personalise the ad experience. These are some of the use cases of graph data structures and applications.

A quick reminder of the definition, graphs are considered to be the composition of vertices (nodes) with respective pairs of edges(also known as links or arcs). In a nutshell, they are viewed to comprise a determined set of nodes and edges which connect these given nodes.

# The rise of eight different types of graph

We are witnessing the rise and adoption of graph databases across different verticals. Gartner acknowledged the five different types of graphs as social, intent, consumption, mobile and interest. In a presentation titled: Graph All the things! Introduction to graph databases, the team from Neo4j captured Gartner’s graph classifications in the illustration below. There is a slight difference in how one of the types of a graph is named by Neo4j in comparison to Gartner. To Neo4j it is a payment graph while to Gartner it is the consumption graph.  There is the argument that a consumption graph is a better name as we do not necessarily pay for every consumption. We will now look at each graph and add some additional types