1. Connected Networks -- 1.0 Introduction -- 1.1 Set Theory -- 1.2 Sets with Two or Less Elements -- 1.3 Generalized Union -- 1.4 Relations and Functions -- 1.5 Superpositions and Inverses -- 1.6 Restrictions -- 1.7 Cartesian Products -- 1.8 Some Special Symbols -- 1.9 Finite Sequences -- 1.10 Networks -- 1.11 Geometrical Realization of a Network -- 1.12 Subnetworks -- 1.13 Degree of a Vertex -- 1.14 Path in a Network -- 1.15 Proper Path in a Network -- 1.16 Reduction of a Path to a Proper Path -- 1.17 Connected Networks -- 1.18 Isolated Vertices -- 1.19 Connected Sets of Branches -- 1.20 Path Connected Set of Branches -- 1.21 Union of Connected Sets of Branches -- 1.22 Connectedness of Paths -- 1.23 Component of a Set of Branches -- 1.24 Existence of Components -- 1.25 Partition into Components -- 1.26 Removal of a Branch -- 2. Loops, Trees, and Cut Sets -- 2.0 Introduction -- 2.1 Loop in a Network -- 2.2 Loops -- 2.3 Subloops of a Loop -- 2.4 Branches and Vertices of a Loop -- 2.5 Paths in a Loop -- 2.6 Removal of a Branch from a Loop -- 2.7 Tree in a Network -- 2.8 Trees -- 2.9 Connected Subset of a Tree -- 2.10 Branches and Vertices of a Tree -- 2.11 Number of Vertices of a Connected Set of Branches -- 2.12 Addition of a Branch to a Tree -- 2.13 Existence of Maximal Trees -- 2.14 Cut Set in a Network -- 2.15 Existence of Cut Sets -- 2.16 Alternate Characterization of Cut Sets -- 3. Incidence Functions and Incidence Matrices -- 3.0 Introduction -- 3.1 Incidence Functions -- 3.2 Matrices and Arrays -- 3.3 Submatrices -- 3.4 Determinants -- 3.5 Incidence Matrices -- 3.6 Square Submatrices of an Incidence Matrix -- 3.7 Unimodular Matrices -- 3.8 Laplacian Expansion of a Determinant -- 3.9 Reduced Incidence Matrix of a Tree -- 3.10 Incidence Matrix of a Loop -- 4. Linear Algebra Review -- 4.0 Introduction -- 4.1 The Field of Scalars -- 4.2 Addition and Scalar Multiplication of Functions -- 4.3 Linear Space of 0-Chains -- 4.4 Canonical Base of the Space of 0-Chains -- 4.5 Inner Product -- 4.6 Linear Maps -- 4.7 Transpose of a Linear Map -- 4.8 Direct Sum Decomposition -- 4.9 Dimension and Direct Sum Decomposition -- 5. Boundary Operator and Coboundary Operator -- 5.0 Introduction -- 5.1 Assumptions of This Chapter -- 5.2 Chain Spaces -- 5.3 The Boundary Operator -- 5.4 Boundaries and Cycles -- 5.5 Summation Over Finite Sets -- 5.6 The Coboundary Operator -- 5.7 Coboundaries and Cocycles -- 5.8 Boundaries, Coboundaries, and Inner Products -- 5.9 Orthogonality of Cycles and Coboundaries -- 5.10 Orthogonality of Boundaries and Cocycles -- 5.11 Decomposition of ?(K) into Cycles and Coboundaries -- 5.12 Decomposition of ? (V) into Boundaries and Cocycles -- 5.13 Isomorphism of Coboundaries and Boundaries -- 5.14 Dimension of the Space of Cocycles -- 6. Axioms of Network Analysis -- 6.0 Introduction -- 6.1 Assumptions of This Chapter -- 6.2 Resistive Networks -- 6.3 Currents and Voltages -- 6.4 Ohm's Law -- 6.5 Sources -- 6.6 Kirchhoff's Laws for Voltage Sources -- 6.7 Kirchhoff's Laws for Current Sources -- 7. Existence and Uniqueness of Solutions -- 7.0 Introduction -- 7.1 Assumptions of This Chapter -- 7.2 Linearity of L and H -- 7.3 Existence and Uniqueness with Voltage Sourcess -- 7.4 Existence and Uniqueness with Current Sources -- 7.5 Current Variables -- 7.6 Voltage Variables -- 8. Kirchhoff's Third and Fourth Laws -- 8.0 Introduction -- 8.1 Assumptions of This Chapter -- 8.2 The Cycle Map -- 8.3 The Chord Map -- 8.4 The Sum of Tree Chord Products -- 8.5 The Current Chain with Voltage Sources -- 8.6 The Coboundary Map -- 8.7 The Tree Branch Map -- 8.8 The Sum of Tree Branch Products -- 8.9 The Voltage Chain with Current Sources -- 8.10 Invariance Under Change of Incidence -- References.