Introduction to Graph Theory authored by Dipak Kumar Ghosh and published by NCBA.

Name: Introduction to Graph Theory authored by Dipak Kumar Ghosh and published by NCBA. - Booksbp.com
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SKU: 9789352551491

1. Fundamental Concepts & Terminology:-

Basic Definitions: Understanding vertices (nodes), edges, loops, parallel edges, and simple graphs.
Degree Concepts: Vertex degrees, isolated vertices, pendant vertices, and the Handshaking Lemma.
Subgraphs: Isomorphism, vertex-deleted subgraphs, and edge-deleted subgraphs.

2. Paths, Cycles, and Connectivity

Walks and Paths: Definitions of walks, trails, paths, circuits, and cycles.
Graph Connectivity: Connected components, cut-vertices, bridges (cut-edges), and \(k\)-connected graphs.
Special Walks: Complete coverage of Eulerian graphs (Euler walks/circuits) and Hamiltonian graphs (Hamiltonian paths/cycles).

3. Trees and Distance

Properties of Trees: Characterizations of trees, forests, and non-trivial properties.
Distance Metrics: Eccentricity, radius, diameter, and center of a tree.
Spanning Trees: Algorithms to find Minimum Spanning Trees (MST) including Kruskal’s and Prim’s algorithms.

4. Vector Spaces & Matrix Representations of Graphs

Matrix Models: Incidence matrices, adjacency matrices, path matrices, and adjacency list forms.
Vector Spaces: Cut-set subspaces, circuit subspaces, and orthogonal vector spaces related to graphs.

5. Planarity and Planar Graphs

Planar Embeddings: Geometric versus combinatorial graphs, regions, and Euler’s Formula (\(V – E + F = 2\)).
Kuratowski’s Theorem: Non-planar graphs (\(K_{5}\) and \(K_{3,3}\)) and homeomorphic graph reduction.

6. Graph Coloring & Directed Graphs

Coloring Problems: Vertex coloring, edge coloring, chromatic numbers, and map coloring rules (Four-Color Theorem).
Digraphs: Directed paths, connectivity definitions (weakly vs. strongly connected), and tournaments.