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Chapter 1

Graph Matching: An Introduction

Graph theory is a branch of mathematics that deals with graphs which are sets of vertices

(or nodes) represented as V(={v1,v2,,vn}) and the associated set of edges represented by

E(={e1,e2,,ek}), where ei=. Graphs are flexible structures that can be used to model

real world entities as well as processes in different domains. The problems in various

domains can be modeled as analogous problems on graphs and solutions to these graph

problems give the solutions to the original problems. This has resulted in widespread use of

graphs in obtaining solutions to problems in various domains. Hence, graph theory has

grown into a significant area of research in mathematics with applications in chemistry,

physics, operations research, social science, biology, computer science etc.

The advances in computer technology and increased applications of graphs have spawned a

renewed interest amongst mathematicians and computer scientists in graph theory. Graphs

have, of late gained significance in solving problems in diverse areas because of the ease of

their representation and manipulation, in computers [Narasingh Deo, 2004]. The problem

of graph matching / similarity of graphs/ graph isomorphism have attracted researchers in

mathematics and computer science from the day Graph theory as discipline has started to

gain importance. The importance of Graph theory and Graph Matching in the current

context is brought out in the ensuing section which also emphasizes the need for

addressing the problem.

1.1 Introduction to Graph Theory and Graph Matching

The paper written by Leonhard Euler on the Seven Bridges of Konigsberg and published in

1736 is regarded as the first paper in the history of graph theory [Narasingh Deo, 2004]. In

1878, Sylvester introduced the term Graph in a paper published in the famous scientific

journal Nature, where he draws an analogy between "quantic invariants" and "co-

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variants" of algebra and molecular diagrams and this lead to different applications of Graph

Theory [Sylvester John Joseph, 1878.] The first textbook on graph theory was written

by Dnes Knig, and published in 1936 [Tute W T, 2001]. A later textbook by Frank Harary,

published in 1969, was popular, and enabled mathematicians, chemists, electrical engineers

and social scientists to talk to each other [Harary Frank, 1969]. Today there are innumerable

number of books on Graph Theory that describe the fundamentals and applications.

The subject of graph theory had its beginning in recreational math problems but it has now

grown in to a significant area of mathematical research. Basically graph theory is a study of

graphs which mathematically model pair wise relations between objects. In all the domains

where graphs are employed for modeling, the vertices or nodes; model the objects whereas

the edges model the relationship. Graphs by their inherent characteristics have been found

to be versatile tools for applications in science and engineering [Narasingh Deo, 2004]. Their

flexibility and robustness in modeling various scenarios and concepts has led to their

popularity. The edges in the graphs may be undirected, representing bidirectional

relationship or they may be directional representing unidirectional relationship. Graphs

which have only undirected edges are referred to as Undirected Graphs, whereas the

graphs which consist of directed edges only are Directed Graphs or Digraphs. Further

graphs which consist of both the types of edges are Mixed Graphs. Graphs are also

categorized as Simple Graphs and Multi Graphs. Graphs which do not have self loops

and parallel edges are simple graphs [Chartrand, 2012]. The graphs having self loops and /

or parallel edges are multi graphs. The various definitions and concepts about graphs are

brought out in section 1.1.1.

1.1.1 Definitions and Concepts

This section provides the necessary definitions and introduces the conventions followed in

the thesis.

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Graph : A graph is an ordered pair G = (V, E) comprising of a set V of vertices or

nodes together with a set E of edges or lines/arcs. The edges may be

directed (asymmetric) or undirected (symmetric).

Vertex set : The set of vertices in a graph is denoted by V (G) or V

Edge set : The set of edges in a graph is denoted by E (G) or E

Degree : The degree of a vertex is the number of edges that are incident to it.

Size : A graph's size is |E|, the number of edges

Order : The order of a graph is |V|, the number of vertices

Distance : The distance d (x, y) in G between two vertices x, y is the length of a

shortest x y path in G

Eccentricity : The eccentricity, e(x), of the vertex x is the maximum value of d(x, y),

where y is allowed to range over all of the vertices of the graph

OR

The eccentricity, e(x), of the vertex x in a graph G is the distance from x

to the vertex farthest from x i.e. e (x) = max d (x, xi), i

Diameter : The greatest distance between any two vertices in G is the diameter of

G, denoted by diam(G)

Center : A vertex with minimum eccentricity in a graph G is referred to as the

center of graph, such a vertex is also called Central Vertex

Radius : The greatest distance between the central vertex and any other vertex

is the radius of the graph and is denoted rad G. It should be obvious that

rad G diam G 2*radG

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Complete

Graph / clique

If all vertices of G are pair wise adjacent, then G is a complete graph or

clique

Walk : A walk is an alternating sequence of vertices and edges, with each edge

being incident to the vertices immediately preceding and succeeding it

in the sequence. A walk of length k is a non-empty alternating sequence

of k+1 vertices and k edges in G

Trail : A trail is a walk with no repeated edges

Path : A path is a walk with no repeated vertices

Closed Walk : A walk is closed if the initial vertex is also the terminal vertex

Cycle : A cycle is a closed trail with at least one edge and with no repeated

vertices except that the initial vertex is also the terminal vertex

Length of a walk: The length of a walk is the number of edges in the sequence defining

the walk

Connected

Graph

A non-empty graph G is called connected if any two of its vertices are

linked by a path in G. A connected graph is a graph with exactly one

connected component

Undirected

Graph

An undirected graph is one in which all edges have no orientation.

Directed graph : A directed graph or digraph is an ordered pair D = (V, A) where each

edge has a direction

:

:

:

:

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Adjacent : Two vertices x, y of G are adjacent if is an edge in G in other

words two vertices are adjacent if they are incident to a common edge.

Similarly two edges are adjacent if they are incident to a common vertex

Incident : A vertex v is incident with an edge e if v e that is e is an edge at v

Independent : A set of vertices or edges are independent if no two of its elements are

adjacent

Isomorphic : G1 and G2 are isomorphic, if there exist a bijection : V1 V2 such

that E1 E2 x, y in V1

Invariant : A mapping taking graphs/ graph parameters as arguments is called a

graph invariant if it assigns equal values to isomorphic graphs

A simple graph : A simple graph is a triple G= (V,E,I), where V and E are disjoint finite sets

and I is an incidence relation such that every element of E is incident

with exactly two distinct elements of V and no two elements of E are

incident to the same pair of elements of V

Connectivity : A graph G has connectivity k if G is k-connected but not (k+1)-

connected. A complete graph on k+1 vertices is defined to have

connectivity k.

Neighbours : The set of neighbours, N(v), of a vertex v is the set of vertices which are

adjacent to v. The degree of a vertex is also the Cardinality of its

neighbour set.

Induced Sub-

Graph :

For a set of vertices X, we use G to denote the induced sub

graph of G whose vertex set is X and whose edge set is the subset

of E(G) consisting of those edges with both ends in X

:

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These are some of the definitions in graph theory that will be used in this dissertation.

Many efficient representation techniques are available for representing graphs in

computers and are briefly described in the next section.

1.1.2 Computer Representation and Graph Spectra

The availability of robust computer representation scheme for graphs and flexibility of

processing them has furthered the use and applications of graphs. The graphs are

represented by various matrices such as adjacency matrix; incidence matrix,