Linear graph theory. Basic definitions of linear graphs

Linear graph theory Linear graph theory, a branch of combinatorial mathematics has proved to be a useful tool for the study of large or complex systems. Leonhard Euler wrote perhaps the first paper on graph theory and laid the foundation for the theory when he published the solution to the Königsberg bridge problem in 1736. However its first application to a problem of physical science did not arise until 1847, when Gustav Kirchhoff applied it to the study of electrical networks. The material presented in this chapter is based mainly on two sources, which I think present a thorough exposition of the theory in which this thesis works is founded. They are the seminal work by Branin [Branin, 1966 #1] on network analogies for physical systems, and the work by Seshu and Reed [Seshu, 1961 #7] on electrical networks. Basic definitions of linear graphs Since the emphasis of this thesis is on applications of graph-theoretic concepts to the problem of system modeling rather than the development of an abstract theory, the basic definitions are stated in a semiformal manner. The basic elements in forming a linear graph are a set of line segments (called an edge) each having two end points (called a vertex) If a direction is specified on each edge, they become oriented edges. Schematically, we can indicate the edge orientation in either of two ways; by attaching a + and a to the two ends of the edge or by attaching an arrowhead directed from the + end to the end of the edge. Definition Linear graph.alinear graph G is a collection of edges, no two of which have a point in common that is not a vertex. A linear graph is also referred to as a 1-complex which is a collection of 1- and 0-simplex. A graph as defined here is an abstract mathematical object and need not have any geometric significance whatsoever. One can regard a 1-complex as a representation of some portion of the 3-dimensional euclidean space, where the 0-simplex represent points in the space. However until we assign vector-values to the 0-simplex, do we get a geometric interpretation of the 1-complex. In this context, the collection of edges is assumed to be finite, thereby resulting in a finite graph. Definition Subgraph.Asubgraph G s is a subset of edges of the graph G. Subgraph G s is a proper subgraph if it does not contain all edges of G. Definition Incidence. Edge k is incident to a vertex p if p is an endpoint of k. Definition Degree of vertex. The degree of a vertex is the number of edges incident to that vertex. Definition Edge train.anedge train is any subset E s of the edges of G where E s can be ordered such that an edge k in E s has a vertex in common with the preceding edge k 1, and the other vertex in common with the succeeding edge k + 1. Definition Path.Apath is an edge train where each internal vertex has degree of exactly two and

a e 1 2 4 5 7 8 9 d 6 c 3 b f FIGURE 1. A directed linear graph each terminal vertex has degree of 1. Definition Circuit or loop.acircuit or loop is a closed edge train where all vertices are of degree 2. It is important to note that a loop is defined as a subgraph rather than as a sequence of vertices as it is customary done. Definition Connected graph. A graph G is connected if there exists a path between any two vertices of the graph. If the graph is not connected, it contains p connected components. A frequently used concept in graph theory is that of a tree. Definition Tree. A tree T is a connected subgraph of a connected graph G which contains all the vertices of G but contains no loops. Definition Free tree. A free tree T is an connected acyclic subgraph of an undirected connected graph G having no defined root vertex. The edges that are not part of the tree form a subgraph T called cotree. Edges in the tree are referred to as branches while edges in the cotree are referred to as links or chords. An important property of any tree is that it contains exactly v 1 edges. If the graph G is not connected, a tree does not exists since by definition a tree is a connected subgraph of a connected graph, however, a tree can be found for each for the connected components of G. The collection of such trees is called a forest. Similarly, each component of G defines a cotree and the collection of all cotrees is called a coforest. It has already been stated that the number of branches in a tree T of a graph G having v vertices and e edges equals v 1 ; consequently the number of chords in the cotree equals e ( v 1) e v+ 1. Further, in a graph with v vertices, e edges and p connected components there will be v p branches in the p trees and e v + p chords in the p cotrees. NEED TO give some examples with reference to the figure above.

3 b f h c 2 g 4 a i d 1 e 5 FIGURE 2. A connected graph with a tree T indicated by heavy lines An important property of trees is that adding any chord between any two vertices in the tree, establishes a circuit. EXAMPLE. Since in a connected graph there are e v+ 1 chords for a given tree T, there are as many unique circuits defined by the chords of T. Definition f-circuits. f-circuits (fundamental circuits) of a connected graph G for a tree T are the e v+ 1 circuits formed by each chord of the given tree T. Consider for example the graph G shown in the figure bellow. The heavy lines indicate a tree T. The number of edges and vertices is 9 and 5 respectively, therefore the number of chords is 5. The five fundamental circuits defined by these chords and their tree paths are given in the table below. TABLE 1. Fundamental circuits for the tree of figure Fig. 2 Chord Tree path f-circuit (f) (a, b) (a, b, f) (h) (b, g, d) (b, g, d, h) (c) (b, g) (b, g, c) (i) (a, g) (a, g, i) (e) (a, g, d) (a, g, d, e) The table includes the f-circuits defined with respect to the tree T (a, b, g, d). It has been proved that any circuit in the graph can be made an f-circuit with respect to some tree [linear graphs]. This property is of utmost importance when we analyze the algebraic structure of the graph since it allows us to select the causality of the model of the physical system being modeled. Since the graph is directed, it is appropriate to consider the f-circuits to be oriented. As stated above, a chord uniquely defines an f-circuit, therefore, it is natural to assign the orientation of the f-circuit to be consistent with the direction of the defining chord. There is a dual to a circuit in a linear graph. This dual is called cut-set.

Definition cut-set. A cut-set of a connected directed graph G, is defined as a set C of edges of G such that the removal of these edges from G leaves G partitioned in exactly two connected components, provided that no subset of C set has the same property as the set C. Definition Fundamental system of cut-sets. A fundamental system of cut-sets with respect to a tree T of a connected directed graph G is the set of v 1 cut-sets in which each cut-set includes a branch of T. The fundamental cut-set orientation is to agree with the orientation of the defining branch. We leave this section by presenting an interesting property of a fundamental cut-set: if represents a branch of a tree T in a connected graph G, and ( e 2, e 3,, e n ) represents the cut-set defined by e 1, then each of the f-circuits defined by the chords e 2, e 3,, e n includes e 1. e 1 Topological properties of linear graphs As is well known, the connectivity relations of any oriented linear graph can be completely specified by means of the augmented incidence matrix, denoted A whose elements are +1, -1, and 0. Definition Augmented incidence matrix. The augmented incidence matrix A of a directed graph is G with v vertices and e edges is a v e matrix defined by a ij 1 if edge j is incident at vertex i and is oriented away from vertexi 1 if edge j is incident at vertex i and is oriented toward vertex i 0 if edge j is not incident at vertex i The incidence matrix was first introduced by Poincare and it contains information both about the orientation of edges in the graph and how they are joined to form nodes. For the graph in Fig. 1 the incidence matrix is given by A 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 (EQ 1) In general, for a graph with p connected components the incidence matrix is a direct sum. A matrix M is said to be a direct sum of M 1, M 2, M p if for any M k in M no nonzero element of M k lies in a row or column of M associated with any of the other submatrices. Since each column of A contains both a single +1 and a single 1, its rows are linearly dependent. Removing any row from A will leave v 1 linearly independent rows. We call this matrix the incidence matrix, denoted A. From a well known theorem from graph theory [linear graphs], we

know that if T is a tree of a connected graph G, the v 1 columns of A that correspond to the branches of the tree T constitute a nonsingular matrix. Thus if a tree is chosen and the columns of A properly arranged, the matrix A can be partitioned into the submatrix A T, referring to the branches of the tree only, and the submatrix A C, referring to the chords or to the cotree. A A T A C (EQ 2) Two new matrices can be defined to describe the topology of the graph. The fundamental circuit matrix (designated B) captures the connectivity relations between circuits and edges. And the fundamental cut-set matrix (henceforth referred to as the cut-set matrix) designated Q. Matrix Q captures the connectivity between cut-sets and edges. Definition Fundamental circuit matrix. The fundamental circuit matrix B of a directed graph G with respect to a tree T are the e v+ 1 circuits formed by each chord. It is defined by b ij 1 if edge j is in circuit i and the orientation of the circuit and the edge conicide 1if edge j is in circuit i and their orientations do not coincide 0 if edge j is not in circuit i If the matrix B is properly arranged it can be partitioned into the submatrix B T referring to the branches of the tree and submatrix B C referring to the chords of the cotree. However, since each chord appears exactly once in any given f-circuit in the positive sense, the matrix B C U C ; a unit matrix. Then B B T U C (EQ 3) A closer look to the matrices A and B will reveal a very interesting and fundamental property of a linear graph. We can define two linear vector spaces associated with the graph G. The vector space V Q spanned by the rows of matrix A and the vector space V B spanned by the rows of matrix B. These two vector spaces are subspaces of the linear vector space V G of dimension e. It can be shown [Seshu, 1961 #7] that the matrices A and B satisfy the following relation: AB T 0 and/or BA T 0 (EQ 4) which implies that the two vector subspaces V Q and V B are orthogonal complements of the e- dimensional linear vector space V G. This fact is known as the orthogonality principle. Using this fact, we can derive an equation to find matrix B T : A T A C B T t A T B t T + A C U C 0 U C (EQ 5) It follows that

a + 1 + + 2 3 c 4 5 + + + + 7 8 d a 1 2 4 3 7 c 5 d 8 b 6 + e b 6 e FIGURE 3. An electrical network and its associated linear graph B T A t C A 1 T ( ) t (EQ 6) When the columns of matrix A are properly arranged such that the first v 1 columns of A are in direct correspondence with the branches of some tree T of a graph G, an equivalent matrix Q can be derived from A. Such matrix (called cut-set matrix) is found by applying row operations to A. Matrix Q represents the fundamental system of cut-sets with respect to the tree T, and includes the v 1 cut-sets of G in which each cut-set includes only one branch of T. Then Q U T Q C (EQ 7) Since matrices A and Q are equivalent, the following relation holds BQ t 0 (EQ 8) It follows from Eq. (5) that B T Q t C (EQ 9) The algebraic structure associated with a linear graph To show the algebraic structure associated with a linear graph, we recur to the following example, which should be common to most readers: the analysis of a simple electrical network. Fig. 3 shows an electrical network and a linear graph topologically equivalent to the network. The numbering of the elements in the network and the edges in the linear graph makes it easy to see the correspondence between them. In the figure all current directions and voltage references have been indicated. It is a trivial exercise to derive the Kirchhoff s current and voltage equations for this network and they are given as follows

i 1 () t + i 2 () t + i 3 () t i 7 () t 0 i 1 () t i 4 () t i 6 () t 0 i 2 () t i 3 () t + i 4 () t i 5 () t 0 i 5 () t + i 7 () t + i 8 () t 0 i 6 () t i 8 () t 0 (EQ 10) and v 1 () t + v 2 () t + v 4 () t 0 v 1 () t + v 3 () t + v 4 () t 0 v 4 () t + v 5 () t v 6 () t v 8 () t 0 v 1 () t v 6 () t + v 7 () t v 8 () t 0 (EQ 11) Rewriting in matrix form the two systems of equations (Eq. (10) and Eq. (11)) we obtain i 1 () t 1 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 i 2 () t i 3 () t i 4 () t i 5 () t i 6 () t i 7 () t i 8 () t 0 (EQ 12) and v 1 () t v 2 () t 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 1 v 3 () t v 4 () t v 5 () t v 6 () t v 7 () t v 8 () t 0 (EQ 13) The coefficient matrices in the previous equations can be recognized as the augmented incidence matrix and the fundamental circuit matrix respectively of the directed graph of Fig. 3. Now it should be noted that this observation is perfectly general and it is applicable to any system for which a directed linear graph can be obtained. This facts lead to the definition of the two Kirchhoff postulates that state that the sum of currents leaving a node equals 0 and the sum of voltages around a loop equals 0. This can be stated by

Ai() t 0 (EQ 14) where A is the augmented incidence matrix of the directed graph and i() t i 1 ()i t 2 () i t e () t where i j () t is associated with edge j. Similarly, Bv 0 (EQ 15) where B is the fundamental circuit matrix of the directed graph with respect to some tree T, and v() t v 1 ()v t 2 () v t e () t where v j () t is associated with edge j. For the example above, the tree T (1, 4, 6, 8). We know that the augmented incidence matrix is a singular matrix for which we can remove a row to obtain the incidence matrix A. Thus, in an oriented graph with v vertices, there are exactly v 1 linearly independent Kirchhoff s current equations. In general if the graph contains p connected components, there are v p linearly independent Kirchhoff s current equations. Similarly, there are e v+ p linearly independent Kirchhoff s voltage equations for a network of p connected components. Theorem I. If T is any tree of a connected graph, the voltage functions of the chords of T can be expressed as linear combinations of the voltage functions of the branches of T, and the current functions of the branches of T can be expressed as linear combinations of the current of the chords of T. Proof. To prove the first part of the theorem let us assume that the columns of B are properly arranged so as to include the chords of the defining cotree as the last e v+ 1 columns, and the vector v is arranged accordingly: B f U v b() t v c () t 0 (EQ 16) Expanding Eq. (16) we obtain B f v b () t + v c () t 0 which can be solved for v c () t : v c () t B f v b () t. This shows that the chord voltages can be expressed as linear combinations of the branch voltages. For the second part of the proof, we recognize that the cut-set matrix Q of v 1 cut-sets and rank v 1 defines a set of equations Qi() t 0 which are equivalent 1 to the Kirchhoff s current equations Ai() t 0. If the columns of Q are properly arranged so as to include the branches of the defining tree as the first v 1 columns, and the vector i is arranged accordingly we have 1. Two systems of linear equations are equivalent if they have the same solution.

UQ f i b () t i c () t 0 (EQ 17) Expanding Eq. (17) we obtain i b () t + Q f i c () t 0 which can be solved for i b () t : i b () t Q f i c () t. This equation defines the branch currents as linear combinations of the chord currents. Equations Eq. (16) are referred to as fundamental circuit equations and equations Eq. (17) are referred to as fundamental cut-set equations.