The Spectral Radii of Some Adhesive Graphs

The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determined.


Introduction
The spectral radius of the graph powerfully characterizes dynamic processes on networks, such as virus spread and synchronization. In [1], the authors pointed out that the maximum eigenvalue of a graph (i.e., the spectral radius) plays an important role in the network virus transmission mode. They found that the small spectral radius and the large robustness suppress the spread of network viruses. At present, a group of excellent experts have used parameters such as maximum degree and girth to give the bounds of the spectral radii of many graphs, see [2]- [11]. In this paper, we will use special methods to give research on the precise spectral radii of some graphs. Let

( )
A G is a real symmetric matrix, as its characteristic roots are all real, and can be sorted as follows: . Multi-sets of n characteristic roots are called the spectrum of graph G. Given an equitable partition ( ) 1 2 , , , k C C C π = of a graph G, we now define the quotient G π of G with respect to π . Let ij C denote the number of How to cite this paper: Wang, Q.N. (2021) The Spectral Radii of Some Adhesive edges which join a fixed vertex in i C to vertices in j C . Then G π is the directed graph with the cells of π as its vertices, and with ij C arcs going from i C to j C . Then say G π a quotient graph of graph G corresponding to partition π .
An outline of the rest of the paper is as follows. In Section 2, we will present important results about quotient graph. In Section 3, we will give the main results. Godsil [12] presented important results about quotient graph as follows.

Main Results
Theorem 1. Let G be a graph obtained by identifying a vertex of k K to every vertex of l C . Then Proof. Checking the structure of graph G, we can obtain an equivalence partition ( ) , where C 1 = {all the vertices on the cycle C 1 }, By Theorem 1, we know that the spectral radius of the bonding graph G is independent of the coil length, only depends on the number of vertices of the complete graph that is bonded.
By Theorem 1, we directly obtain the following result.  ( ) Proof. Checking the structure of graph G, we get an equivalence partition ( ) If G is a graph constituted by splicing two complete graphs of order k on vertex v, see Figure 2.
Assume that G is a prism graph. Then  undersides after cutting off the side edges of the prism with a plane. Then the spectral radius of the resulting graph is ( ) Proof. Checking the structure of G, we get an equivalence partition ( ) If the above pyramid is expanded into a plane, also known as wheel graph.
The undersides (with n vertices) of two identical pyramids are glued together to form a spindle graph, the spectral radius of this graph satisfies the following corollary.
The cone points (with n vertices on the base) of two identical pyramids are glued together to form a dumbbell graph, the spectral radius of this graph satisfies the following corollary. Corollary 7.
( ) The cone points (with n vertices on the base) of two identical pyramids are glued with one edge to form a barbell graph, the spectral radius of this graph satisfies the following corollary. Corollary 8. Proof. According to the structure of graph G, we get an equivalence partition ( ) , Theorem 8. Let G be a graph that obtained by adding m edges with one-to-one correlation between m pairs of vertices of two k-order complete graphs, then C V G C = − . We can construct quotient graph G π of G, we have

Conclusion
Lemma 2 shows that, the above results give an upper bound of the spectral radius of the corresponding subgraph. The key to characterizing the spectral radius of a graph by using the property of quotient graph is to construct equivalence partition. The adjacency matrix of the quotient graph is always smaller than the adjacency matrix of the supergraph, so it is a beautiful way to use the property of quotient graph to depict the spectral radius of graph.