Graphs and Trees. An example. Graphs. Example 2

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Graphs and Trees An example How would you describe this network? What kind of model would you write for it? What kind of information would you expect to obtain?

1 Graphs and Trees An example How would you describe this network? What kind of model would you write for it? What kind of information would you expect to obtain? Relationship between some of the apoptotic genes in homosapiens JGomes, SBS IITD 2 Example 2 Graphs Definition: v 1 Nature, 460,16 (2009) Graphical representation of the current budding yeast interaction network Plectix BioSystems in Somerville, Massachusetts Cellucidate software to model the epidermal growth factor receptor pathway, potential states including all protein complexes and phosphorylation states for the system. Complexity of cell signalling networks What interaction data do they use in their models? JGomes, SBS IITD 3 A graph G is a finite nonempty set V together with an irreflexive, symmetric relation R on V. Since R is symmetric, for each ordered pair (u, v) R, the pair (v, u) also belongs to R. The notation E is used to denote the set of symmetric pairs in R. The number of vertices is called the order of G Each element of E consists of two symmetric ordered pairs from R E is called the edge set of G Hence V = order of G and E = size of G V(G) = R = E(G) = {v 1, v 2, v 3, v } 4 {(v 1, v 2 ), (v( 2, v 1 ), (v( 2, v 3 ), (v( 3, v 2 ), (v( 3, v 1 ), (v( 1, v 3 ), (v( 4, v 1 ) (v 1, v 4 ) } {(v 1, v 2 ), (v( 2, v 3 ), (v( 3, v 1 ), (v( 4, v 1 )} JGomes, SBS IITD 4 v 2 v 4 v 3 1

2 If e = uv E(G), then uv is an edge of the graph G and e joins the vertices u and v. If uv E(G), then u and v are adjacent to or adjacent with each other. If uv E(G), then u and v are nonadjacent vertices If uv and uw are distinct edges of a graph (v w), then uv and uw are adjacent edges graphs v 1 v 2 v 4 v 3 Examples Let n 2 be an integer. If G is a graph of order n, what is the minimum size possible for G ( in terms of n) if G contains a vertex which is adjacent to all other vertices If V is a nonempty set, why does it follow that the empty subset of VxV is an irreflexive, symmetric relation on V? JGomes, SBS IITD 5 JGomes, SBS IITD 6 Networks as Mathematical Models An example v1 Definition: A network is a graph or digraph together with a function which maps the edge set into a set of real numbers. A network resulting from a graph is called an undirected network; a network resulting from a digraph is called a directed network. v2 v3 JGomes, SBS IITD 7 JGomes, SBS IITD 8 2

3 The degree of a vertex Sample Problems There are two numbers associated with graphs, the order and the size. We write this in the form G(p, q) where p denotes the order of the graph (the number of vertices V) and q denotes the size of the graph (the number of edges R). The degree of a vertex deg v is the number of edges of G that is incident on the vertex v. G(1, 0) deg v = 0 G(2, 1) deg v = 1 G(3, 3) deg v = 2 G(5, 10) deg v = 4 1. Verify p i= 1 deg v = i 2 q A contrived metabolic network JGomes, SBS IITD 9 JGomes, SBS IITD 10 Isomorphic graphs Two equal graphs are called isomorphic. An isomorphism from G 1 and G 2 means a one to one mapping φ : V( G1 ) V( G2 ) from V(G 1 ) onto V(G 2 ) such that if v 1 and v 2 are adjacent on G 1, if and only if the points φ(v 1 ) and φ(v 2 ) are adjacent on G 2. Connected and disconnected graphs Two vertices u and v in a graph G are connected if u=v, or if u v and a u v path exists in G. A graph is connected if every two vertices of G are connected; Otherwise G is disconnected. A u v trail in which u=v and which contains at least three edges is called a circuit. A circuit must start and end at the same vertex. A circuit that does not repeat any vertices (except the first and last) is called a cycle. JGomes, SBS IITD 11 JGomes, SBS IITD 12 3

Forests and trees Examples of trees A connected graph that has no cycles is called a tree. It is customary to define a graph without any cycles (be it connected or not) as a forest.

Then there is exactly one u v path in G.

4 Forests and trees Examples of trees A connected graph that has no cycles is called a tree. It is customary to define a graph without any cycles (be it connected or not) as a forest. Hence, each component of a forest is called a tree. The word tree is used because, when drawn, some of them look like trees. Theorem Let u and v be two vertices of a tree G. Then there is exactly one u v path in G. Theorem If G is a tree of order p and size q, then q = p 1 Definition If T is a subgraph of G and T contains every vertex of G, such a tree is called a spanning tree of G. JGomes, SBS IITD 13 A rooted phylogenetic tree, which has been colored according to the threedomain system. The most common method for rooting trees is the use of an uncontroversial outgroup close enough to allow inference from sequence or trait data, but far enough to be a clear outgroup. Tree of Human Races JGomes, SBS IITD 14 Tree of Evolution Delhi Metro Route Map JGomes, SBS IITD 15 JGomes, SBS IITD 16 4

5 Eulerian & Hamiltonian Paths An Eulerian path is a path that traverses each edge in a network A Hamiltonian path traverses each vertex only once JGomes, SBS IITD 17 5

Let G = (V, E) be a graph. If u, v V, then u is adjacent to v if {u, v} E. We also use the notation u v to denote that u is adjacent to v.

Graph Adjacent Endpoint of an edge Incident Neighbors of a vertex Degree of a vertex Theorem Graph relation Order of a graph Size of a graph Maximum and minimum degree Let G = (V, E) be a graph. If u,