The Adjacency Matrix and The nth Eigenvalue
نویسنده
چکیده
In this lecture, I will discuss the adjacency matrix of a graph, and the meaning of its smallest eigenvalue. This corresponds to the largest eigenvalue of the Laplacian, which we will examine as well. We will relate these to bounds on the chromatic numbers of graphs and the sizes of independent sets of vertices in graphs. In particular, we will prove Hoffman’s bound, and some generalizations. Warning: I am going to give an alternative approach to Hoffman’s bound on the chromatic number of a graph in which I use the Laplacian instead of the adjacency matrix. I just worked this out last night, so I still don’t know if it is a good idea or not. But, I’m going to go with it. My proof of Hoffman’s bound in the regular case will be much simpler than the proof that I gave in 2009.
منابع مشابه
The Main Eigenvalues of the Undirected Power Graph of a Group
The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...
متن کاملTHE RELATION BETWEEN TOPOLOGICAL ORDERING AND ADJACENCY MATRIX IN DIGRAPHS
In this paper the properties of node-node adjacency matrix in acyclic digraphs are considered. It is shown that topological ordering and node-node adjacency matrix are closely related. In fact, first the one to one correspondence between upper triangularity of node-node adjacency matrix and existence of directed cycles in digraphs is proved and then with this correspondence other properties of ...
متن کاملStudy of fullerenes by their algebraic properties
The eigenvalues of a graph is the root of its characteristic polynomial. A fullerene F is a 3- connected graphs with entirely 12 pentagonal faces and n/2 -10 hexagonal faces, where n is the number of vertices of F. In this paper we investigate the eigenvalues of a class of fullerene graphs.
متن کاملOn Eccentricity Version of Laplacian Energy of a Graph
The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini an...
متن کاملA note on energy of some graphs
Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy of a graph is the sum of the absolute values of its eigenvalues. In this note we obtain analytic expressions for the energy of two classes of regular graphs.
متن کامل