Matrix Inequalities for Iterative Systems

5.1.1 Bipartite Graphs -- 5.1.2 Forests and Trees -- 5.1.3 Construction of Worst Case Trees -- 5.2 Semiregular Graphs -- 5.3 Trees -- 5.3.1 Stars and Paths -- 5.3.2 Walks of Length 2 and 3 in Trees -- 5.3.3 Walks of Length 4 and 5 in Trees -- 5.3.3.1 (1, n1, n2)-Barbell Graphs (Trees with Diameter 3) -- 5.3.3.2 (2, n1, n2)-Barbell Graphs -- 5.3.3.3 Proof for the w5-Inequality for Trees -- 5.3.4 A Conjecture for Trees -- 5.4 Subdivision Graphs -- 5.4.1 Walks of Length 2 and 3 -- 5.4.2 Walks of Length 4 and 5 -- III: DIRECTED GRAPHS / NONSYMMETRIC MATRICES -- 6: Walks and Alternating Walks in Directed Graphs -- 6.1 Chebyshev's Sum Inequality -- 6.2 Relaxed Inequalities Using Geometric Means -- 6.3 Alternating Matrix Powers and Alternating Walks -- 6.3.1 Definitions and Interpretations -- 6.3.2 A Simple Generalization -- 6.3.3 Related Work -- 6.3.4 Transferring Hermitian Matrix Results to General Matrices -- 6.3.5 Generalizations for Nonnegative Matrices -- 6.3.5.1 Implications for Hermitian Matrices -- 6.3.6 Generalizations for Complex Matrices -- 6.3.6.1 Implications for Hermitian Matrices -- 7: Powers of Row and Column Sums -- 7.1 Degree Powers and Walks in Directed Graphs -- 7.1.1 Related Work -- 7.1.2 Extended Results -- 7.2 Row and Column Sums in Nonnegative Matrices -- 7.3 Other Inequalities for the Number of Walks -- 7.4 Geometric Means -- IV: APPLICATIONS -- 8: Bounds for the Largest Eigenvalue -- 8.1 Matrix Foundations -- 8.1.1 Complex Matrices -- 8.1.2 Normal Matrices -- 8.1.3 Real Symmetric and Hermitian Matrices -- 8.1.4 Nonnegative Matrices -- 8.1.5 Symmetric Nonnegative Matrices -- 8.2 Graphs and Adjacency Matrices -- 8.2.1 Directed Graphs -- 8.2.2 Directed Graphs with Normal Adjacency Matrix -- 8.2.3 Undirected Graphs -- 8.2.4 Other Bounds -- 8.3 Eigenvalue Moments and Energy -- 8.3.1 Energy Bounds -- 8.3.2 Closed Walks Bounds