# Effect on normalized graph Laplacian spectrum by motif attachment and duplication

###### Abstract

To some extent graph evolutionary mechanisms may be explained from the spectrum. Motif (subgraph) duplication and attachment are two relevant graph operations take place in the evolution of real networks. On other hand, a high (algebraic) multiplicity of eigenvalue and others has been observed in the spectrum of many real networks. Here we investigate how the spectrum of the normalized graph Laplacian gets affected by motif duplication and attachment. We describe some results related to the eigenvalue . Generalized results are also described. And they are verified with suitable examples.

AMS classification: 05C75; 47A75

Keywords: Normalized graph Laplacian; Graph spectrum; Eigenvalue 1; Motif doubling; Motif attachment.

## 1 Introduction

Now a days spectral graph theory is playing an important role to analyze the structure of real networks [18, 6, 4]. It has been shown that how the graph evolutions can be described from different eigenvalues which leads to build hypothesis to explain biological and other processes that reform the connectivities in an underlying graph of a network [5, 2]. Duplication of an induced subgraph and coupling (joining) of a (smaller) graph into the existing graph, are two biologically relevant evolutionary porousnesses [2], are interest of our study. Here we intensively investigate the production of eigenvalue 1 and others, which are mostly observed in the real (biological and other) networks [3, 6], by the above mentioned graph operations.

Let be a connected and finite graph of order with the vertex set and the edge set . If two vertices are connected by an edge in , they are called neighbors, . Let be the degree of , that is, the number of neighbors of . For functions we define the normalized graph Laplacian as

(1) |

Note that this operator (studied in [1]) is different from the (algebraic) graph Laplacian operator,
(see [7, 12, 16, 8, 17] for this operator). But our operator (1) is similar to the Laplacian:
investigated in [10] and thus both have the same spectrum.

Suppose be an eigenvalue of . Then is

(2) |

where a nonzero solution is called an eigenfunction corresponding to the eigenvalue . Let be the algebraic multiplicity of the eigenvalues . The eigenvalue equation (2) becomes

(3) |

In particular, when the eigenfunction vanishes at , then , and conversely (for eigenvalue 1 converse is not always true).

For the equation (3) becomes

(4) |

which is a special property of an eigenfunction for the eigenvalue 1 since it doesn’t depend on the degrees of the vertices.

In [10] F. Chung described that,

(5) |

where denotes the adjacency matrix of and (see [9]). Now (5) can be written as,

(6) |

One can see that the equation (6) together with Sylvester’s law of inertia ([15], theorem ) shows that the nullity of equals to the algebraic multiplicity of eigenvalue in . That is,

(7) |

In this paper we show some graph operations which produce eigenvalue with certain algebraic multiplicity. This leads to the constructions of graphs with certain nullity in their respective adjacency matrix. This is another reason for having a special interest on eigenvalue studied here. Thus the next section only contains the results on eigenvalue 1 produced by the graph operations like vertex doubling, motif (an induced subgraph) doubling and graph attachment. In the consecutive section we generalize the results on the production of other eigenvalues by motif duplication and graph joining operations and illustrate with some corollaries and examples.

Most of the results are proved here in geometric way, by finding eigenfunctions corresponding to the eigenvalues.

Now we recall some of the basic properties of the eigenvalues and eigenfunctions of the operator (1) from [14, 7, 1]. The normalized Laplacian is symmetric for the product

(8) |

for real valued functions on . Since , all eigenvalues of are non-negative. If we arrange all the eigenvalues in non-decreasing manner we have

Now a graph is bipartite iff . For a connected graph only the smallest eigenvalue with an constant eigenfunction. Hence, for all other eigenfunctions , from (8) we get

(9) |

For a graph with vertices and the equality holds if and only if the graph is complete. Thus for a complete graph,

(10) |

## 2 Eigenvalue 1

The properties (4) and (7) make the eigenvalue interesting for us. [13] systematically investigated the effect of the addition of a single vertex on . In [1] authors described the effects with more graph operations. Here we investigate the same to generalize the results for a wide range of operations. In fact using graph operations like vertex doubling and motif doubling one can construct a graph with eigenvalue of desire multiplicity.

### 2.1 Vertex Doubling:

Doubling of a vertex of is to add a vertex to and connect it to all in whenever . Vertex doubling of a vertex of ensures the eigenvalue 1, with an eigenfunction , which takes value 1 at , -1 at its double and 0 otherwise [1].