Graph theory is an exciting sub-field in mathematics with numerous applications. For example, graph theory offers highly effective techniques for developing interconnected systems and intricate networks with applications in computer science, data structures, telecommunications, machine & deep learning, networking, utilities, and the like.
The concepts and the mathematics powering graph theory’s potent techniques can seem pretty complicated. So a solid understanding of analytical geometry, linear algebra, number theory, and combinatorics is essential. And loads of practice on different graph theory problems is another crucial requirement for good assignment scores.
This article here is a quick and concise recapitulation of almost every central graph theory concept. So, if your assignments are giving you trouble, this write-up offers some instant graph theory assignment help.
Basic Concepts in Graph Theory
Graph theory explores the behavior of graphs, a set of interconnected points/nodes connected via lines/edges. These points or nodes can be people, computers, mobile towers, etc. Understanding the nature of abstractions and the mathematical representations of such graphical structures is what graph theory dwells into.
Set theory and functions can be used to define the structure of any graph.
- A graph can be considered to be an ordered triple set as follows à{V(G), E(G), ψG}
V(G) is a non-empty set comprising all the vertices & nodes in a graph, E(G) is disjoint with E(G) & includes all edges in the structure, and ψG is an incidence function that relates the edges with an unordered pair of vertices. Here’s an example to elucidate.
Consider the following diagram:
,
V ={A, B, C, D}, E={a, b, c, d, e}
Then
ψG = (a{A,B}, b{A,B}, c{A,C}, d{B,C},e{B,C}, f{B,C}, g{B,D})
- A walk involves traversing a sequence of edges and vertices wherein a particular edge & vertex can be visited more than once. A path is a walk where each vertex can be visited at most once.
- A walk is considered closed if it ends at the starting vertex. A cycle is a closed path involving the edge connecting the starting & the ending vertices.
- A graph is said to be connected there exists a path between each pair of vertices.
- A tree is a connected graph but has no cycles.
- A forest is a graph whose every connected component is a tree.
- The degree of a node in a graph is the number of edges it connects to.
- The size of a graph is the total number of vertices it’s made of.
The above definitions are the basis of every graph theory problem and numerical. If the above seems alien, put in some effort and, if need be, look for professional graph theory assignment help from experts.
Next up are some significant postulates in graph theory.
Claims & Postulates Under Graph Theory
- A finite tree must have at least two leaves. And a tree with size n has (n-1) edges.
- If a graph G(V, E) exhibits any of the following features, it will exhibit the third one too.
- G is connected,
- G has no cycles, and
- |E|= |V|-1
Graphs showcasing any of the two properties above is a tree.
- If we conduct a closed walk on a graph, G(V, E) is known as anEulerian Circuit, only if the walk traverses every edge in E only one.
This forms the basis of Euler’s theorem à A G(V, E) is Eulerian if and only if G is connected ( except any isolated vertices) and the degree of every vertex is even.
- Another postulate involving Eulerian Circuits states that a graph will be an Eulerian circuit iffy is connected. For all vertices in V, indegree(v) is equal to outdegree(v).
- Spanning Trees and Minimum Spanning Trees are vital concepts in graph theory with significant applications.
Suppose we have a connected graph G (V, E) where traversing each edge exacts a cost of c(e)>0. If the cost of spanning the entire graph is Σe ε E c(e), then the spanning tree is a subgraph of G, which is also a tree with the vertices of the vertex set V and edge (V-1).
And the minimum spanning tree of G is the one that has the minimum cost.
Kruskal’s Minimum Spanning Tree Greedy Algorithm
- Sort all the edges in a graph in increasing order of their weight.
- Choose the edge with minimal weight or cost. Next, determine whether it forms a cycle with the spanning tree designed so far. If a cycle is not formed, include that edge; otherwise, discard it.
- Keep on repeating step 2 until there are (V-1) edges in the spanning tree
The above algorithm is called greedy since it involves choosing the edge with the most negligible weight that does not result in a cycle in the constructed spanning tree.
Kruskal’s algorithm finds applications in television networks, LANs, electric grids, pipeline & cable networks, etc.
Let’s wrap up this write-up with an example showcasing the application of Kruskal’s Greedy Algorithm of MST.
A Sum On Kruskal’s Greedy MST Algorithm
Consider the following graph.
The graph contains 9 vertices and 14 edges. The MST must be made of 9 vertices and 8 edges.
After sorting all edges, we have:
| Weight | Source | Destination |
| 1 | 7 | 6 |
| 2 | 8 | 2 |
| 2 | 6 | 5 |
| 4 | 0 | 1 |
| 4 | 2 | 5 |
| 6 | 8 | 6 |
| 7 | 2 | 3 |
| 7 | 7 | 8 |
| 8 | 0 | 7 |
| 8 | 1 | 2 |
| 9 | 3 | 4 |
| 10 | 5 | 4 |
| 11 | 1 | 7 |
| 14 | 3 | 5 |
Next, we pick the edges as per the sorted list. Finally, pick edges that do not form any cycle and stop when the number of edges in the MST equals (V-1).
Try to solve this MST problem using the table above. But, if it seems too difficult, you will have trouble solving your assignments. Look for online graph theory assignment help from graph theory assignment experts urgently.
Another powerful and popular greedy MST algorithm & a counterpart of Kruskal’s algorithm is PRIM’s algorithm.
Key Points Of Prim’s Algorithm
- Prim’s algorithm begins with an empty spanning tree.
- The prime idea behind Prim’s algorithm involves maintaining two particular sets of vertices. The first set has all the vertices within the MST, while the other has the ones not yet included.
- The algorithm determines which edges connect two sets at each step and then picks the edge with the minimum weight. After choosing the edge, that edge’s other endpoint is added to the set of vertices included in the MST.
So, the basic idea behind Prim’s Algorithm is that spanning trees must be connected; thus, the two disjoint sets of vertices must also be connected. And they are to be connected with edges of minimum weight.
That’s it for today. Here’s hoping this article acts as a quick graph theory assignment help guide for one and all.
Graph theory is a challenging sub-domain in advanced and applied mathematics. If the laws, postulates, and problems seem too complex, you will need the help of skilled & professional graph theory assignment experts.
All the best!
Author-Bio: Amily Parker is a computer science and applied mathematics professor from a major public university in Texas, the USA. She also provides algebra homework help as a tutor at MyAssignmenthelp.com, a leading global academic service provider.
