Lecture 1 - Warm-up with graphs
Lecture 2 - Incidence matrix, cofactors of the Laplacian matrix (towards Matrix-Tree Theorem)
Lecture 3 - Determinant of submatrices of incidence matrices and characterization of spanning trees
Lecture 4 - Cauchy-Binet theorem and the proof of matrix-tree theorem
Lecture 5 - Rank of incidence matrix and the number of connected components
Lecture 6 - Rank and acyclic graphs, 0-1 incidence matrix and characterization of bipartite graph
Lecture 7 - Introduction of the eigenvalues and eigenvector
Lecture 8 - Adjacency matrix and number of walks; powers of a symmetric matrix
Lecture 9 - The spectral theorem
Lecture 10 - Rayleigh quotient and the eigenvalues
Lecture 11 - A bound on the diameter of a graph using the number of distinct eigenvalues
Lecture 12 - The rank of a symmetric matrix equals the number of nonzero eigenvalues
Lecture 13 - Perron-Frobenius theorem for nonnegative symmetric matrices
Lecture 14 - The consequences of the Perron-Frobenius theorem (the properties of the largest eigenvalue and the corresponding eigenvectors)
Lecture 15 - Eigenvalues of a graph and its subgraph; a relation between the largest eigenvalue and the degree(s)
Lecture 16 - Cauchy interlace theorem
Lecture 17 - Bipartite graphs and their eigenvalues
Lecture 18 - Positive semi-definite matrices and the eigenvalues of their partitioned form
Lecture 19 - The eigenvalues of a symmetric matrix in partitioned form
Lecture 20 - Bounds on chromatic number, clique number, and independence number
Lecture 21 - Introduction to lazy random walk, walk matrix
Lecture 22 - Convergence of lazy random walk
Lecture 23 - Rate of convergence of lazy random walk
Lecture 24 - Regular graphs, spectral properties, bound on size of independent set
Lecture 25 - Strongly regular graphs, their eigenvalues, and characterization
Lecture 26 - The friendship theorem
Lecture 27 - Introduction to cospectral graphs
Lecture 28 - Godsil McKay switching to generate cospectral graphs
Lecture 29 - Cartesian product and their eigenvalues, Shrikhande graph
Lecture 30 - Graph determined by the spectrum
Lecture 31 - Sensitivity, block sensitivity and hypercube graph
Lecture 32 - Proof of sensitivity conjecture
Lecture 33 - Introduction to expander graphs
Lecture 34 - A lower bound on Cheeger constant, and expander mixing lemma
Lecture 35 - Introduction to Ramanujan graphs
Lecture 36 - 2-Lift and Zig-zag product for construction of expander and Ramanujan graphs
Lecture 37 - Introduction of algebraic connectivity and Fiedler vector
Lecture 38 - Upper bound on algebraic connectivity
Lecture 39 - Some more bounds on algebraic connectivity
Lecture 40 - Relation between algebraic connectivity of graph and its induced subgraph
Lecture 41 - Introduction to Balanced Signed Graphs
Lecture 42 - Sign Switching in Signed Graphs
Lecture 43 - Laplacian of Signed Graphs, Frustration number and Frustration index
Lecture 44 - Determinant, Permanent in terms of graph structure
