Incidence Matrices of X-Labeled Graphs and an Application

In this work, we have made some modifications to the definition of the incidence matrices of a directed graph, to make the incidence matrices more suitable for X – Labeled graphs. The new incidence matrices are called the incidence matrices of X – Labeled graphs, and we have used the new definition to give a computer program for Nickolas` Algorithm .


Introduction
n Abdu (1999), Nickolas described an algorithm to change two core graphs of one type of branch points to core graphs with two or more types of branch points.Since incidence matrices of directed graphs do not deal with labeled graphs, from this point we worked to make the incidence matrices to be more suitable for X -Labeled graphs.This work is divided into three sections.In section 1, we give basic concepts about free groups and graphs.In section 2, we give the definition of the incidence matrices of X -Labeled graphs, and some definitions and results on incidence matrices of X -Labeled graphs.In section 3, we apply the concept of incidence matrices of X -Labeled graphs to give a computer program for Nickolas` Algorithm.

Basic concepts:
Let F be a group and X be a subset of F .Then we say that F is a free group on X , if for any group G and any mapping and e e = .A subgraph Ω of a graph Γ is a graph with ( ) ) (e t Ω and e have the same meaning in Γ as they do in Ω .If Ω ≠ Γ , then we call Ω a proper subgraph.A component of a connected graph Γ is a maximal connected subgraph of Γ .The number of edges incident with the vertex v is called the degree of the vertex v and denoted by ( ) . Now let F be a group and X be a subset of F .Then the graph ( , ) such that the initial vertex of the edge ( ) and x is the labeled of the edge ( ) , and the number of edges in the spanning tree of

Incidence matrices of X -Labeled graphs
In this section we will give the definition of incidence matrices of X -Labeled graphs and some definitions and results related to it.As we know there are two types of matrices to describe graphs, which are called adjacency (or vertex incidence) matrices and incidence matrices.Recall that the incidence matrices of directed graphs Γ are without loops and with n vertices and m edges ( i.e. it is and the incidence matrices of directed graphs do not deal with the labels of edges, so we will put more conditions on the incidence matrices of directed graphs as below.
and the X -Labeled graph Γ has loops with labeling a or b then choose a mid point on all edges labeled a or b to make all of them two edges labeled aa or bb respectively.Therefore in the rest of this work we will assume that all X -Labeled graphs Γ are without loops.
. The starting row of a scale . Proof: By the definition of the product of core graphs and definition 3.3 the result follows.2) The representation of Nickolas` Algorithm preserves the number of branch rows in )) ( ( H M X * Γ , because it is applied on non -branch rows, so apply steps I, II and III reduce the non -branch rows only and then the number of branch rows in )) ( ( H M X * Γ will be still the same as before and the type of branch rows may change only. 3)The representation of Nickolas` Algorithm preserves the number of branch rows in types of branch points as follows: b -sources b -sinks a -sources a -sinks matrix of X -Labeled graph Γ with n rows and m columns, then incidence matrix of X -Labeled graph Γ with r rows and c type b -sources, say , then there are at least n rows with only two non -zero entries rows and all of them are of one type b - a x ij = , and the scale of type 2 S ending with the row only one type of branch points , b -sources,say, into new core graphs with two or more types of branch points.Therefore we will represent Nickolas` algorithm in the form of incidence matrices of X -Labeled graphs in order to give a computer program to change the type of branch points of core graphs with only one type of branch points into two or more types of branch points. of one type, b -sources, say, then we will use the representation of Nickolas` algorithm to change more of branch rows, after k -times .The steps are given below: 0) Delete all zero columns and zero rows if they appear;) 0 * If the branch rows are not of one type, then stop.Otherwise, change the non -zero entries to make all branch rows of type b -sources by reversing the labeling of the columns and then proceed to step 1; 1 ) If i r is the ending and the starting row of the columns j c and

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new row t r ′ , and if there is no such a row then return to step 0 above.Note: In the program of the above Algorithm we will consider 1 − a and 1 − b as a − and b − respectively and then will represent a and b by 1 and 2 respectively.The representation of Nickolas` Algorithm in the form of incidence matrices of X -Labeled graphs is well defined, which means the representation of Nickolas`Algorithm satisfying the following conditions:1) At each step of the representation of Nickolas` Algorithm, we get consistent incidence matrices of X -Labeled graphs because step I is applied to non -branch rows i r and t r to give a non -branch row t r I always gives consistent incidence matrices of X -Labeled graphs .Also step II is applied on non -branch rows of )= , and the row t r which is either a non -branch row with nonor a x th = , or t r is a branch row of type b -sources , so when we apply step III either we have Therefore step III always gives consistent incidence matrices of X -Labeled graphs.
Two free groups F and K are called isomorphic if and only if F and K have the same rank., we say that the edge e joins the vertex ( ) i e to the vertex ( ) t e .The vertex ( ) i e is called the initial vertex of e and ( ) t e is called the terminal vertex of e .Moreover for each e in ( )

we can reduce the degree of the vertices into vertices of degree 2 and 3 only, by isomorphic embedding of F into a free group K on { }
Hw to Hwx .It is also denoted by( ) H is called the cyclomatic number and the cyclomatic number of * Γ ( ) H is the minimal number of edges that we can delete to make a tree.The rank of the finitely generated subgroup H of a free group on X is the cyclomatic number of *Γ ( )H and denoted by ( ) r H . Definition 1.1 A consistent graph is a directed X -Labeled graph Γ on is incident with at least three distinct columns j c , h c and k c at the non -zero entries, then the row i r is called a branch row.If the row i r is incident with only one column j c at the non zero entry and all other entries of i r are zero, then the row i r is called isolated row.A scale in r other entries of i