A nice graph theory book with examples written for SageMath by David Joyner, Minh Van Nguyen, David Phillips.
The book is under the following license http://www.gnu.org/copyleft/fdl.html.

Graph theory is fundamental in many fields.
You do not need to understand everything of it, you need to know which tool exists and where/when to use it.

Table of content:

Introduction to graph theory
    Graphs and digraphs
    Subgraphs and other graph types
    Representing graphs in a computer
    Graph transformation
    Isomorphic graphs
    New graphs from old
Trees and forests
    Properties of trees
    Minimum spanning trees
    Binary trees
    Huffman codes
    Tree traversals
Shortest paths algorithms
    Distance labels
    Searching graphs
    Bellman-Ford algorithm
    Dijkstra's algorithm
    Topological sort
    All-pairs shortest paths
Graph data structures
    Priority queues
    Binary heaps
    Binomial heaps
    Binary search trees
Distance and connectivity
    Path and distance
    Vertex and edge connectivity
    Menger’s theorem
    Network reliability
Centrality and prestige
    Vertex centrality
    Edge centrality
    Ranking web pages
    Hub and authority
Optimal graph traversals
    Eulerian graphs
    Hamiltonian graphs
    Chinese postman problem
    Traveling salesman problem
Graph coloring
    Vertex coloring
    Edge coloring
    Chromatic polynomial
    Assignment and scheduling
Maximum flow problems
    Flows and cuts
    Ford-Fulkerson theorem
    Edmonds-Karp algorithm
    Goldberg-Tarjan algorithm
    Path-flow decomposition
    Maximum weight closure
Algebraic graph theory
    Laplacian and adjacency matrices
    Eigenvalues and eigenvectors
    Algebraic connectivity
    Graph invariants
    Cycle and cut spaces
Random networks
    Network statistics
    Binomial random networks
    Erdos-Renyi networks
    Small-world networks
    Scale-free networks

Follows the link down below of the lastest revision:

https://storage.googleapis.com/google-code-archive-downloads/v2/code.google.com/graphbook/latest-r1991.pdf

Comment

There is some additional materials for this book here: https://bitbucket.org/mvngu/graphbook-supplement

SageMath (http://www.sagemath.org/index.html) seems to be an interesting project:
SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers.
Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.