On the Generalized Adjacency Spread of a Graph

Graphs with maximal signless Laplacian spectral radius

Graph theory plays a fundamental role in various fields of science and engineering, providing powerful tools for modeling and analyzing relationships among entities. One of the most effective ways to study graphs is through matrix representation. This paper explores the three primary matrix representations of graphs: the adjacency matrix, The adjacency matrix provides direct insight into vertex connectivity and the incidence matrix, the incidence matrix reflects the relationship between edges and vertices. and the Laplacian matrix defined as the difference between the degree matrix and the adjacency matrix, plays a central role in spectral graph theory. Matrix representations enable efficient storage, computation, and analysis of graphs using linear algebraic techniques. They form the basis for many modern algorithms in graph theory, This paper discusses the mathematical foundations, construction methods, and practical applications of these matrix forms, highlighting their essential role in both theoretical and applied graph analysis.

The maximum eigenvalue of the adjacency matrix (AM) has been supposed to contain rich information about the corresponding network. An experimental study focused on revealing the meaning and application of the maximum eigenvalue is missing. To this end, AM was constructed using mutual information (MI) to determine the functional connectivity with electroencephalogram (EEG) data recorded with a mental fatigue model, and then was converted into both binary and weighted brain functional network (BFN) and corresponding random networks (RNs). Both maximum eigenvalue and corresponding network characters in BFNs and RNs were considered to explore the changes during the formation of mental fatigue. The results indicated that large maximum eigenvalue means more edges in the corresponding network, along with a high degree and a short characteristic path length both in weighted and binary BFNs. Interestingly, the maximum eigenvalue of AM was always a little larger than that of the corresponding random matrix (RM), and had an obvious linearity with the sum of the AM elements, indicating that the maximum eigenvalue can be able to distinguish the network structures which have the same mean degree. What is more, the maximum eigenvalue, which increased with the deepening of mental fatigue, can become a good indicator for mental fatigue estimation.

Possible energy spectra of systems constructed from finite bethe lattices

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (In + L(G))−1= ω(G) = (wij). If, for example, w(G) = min wij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = Kn. If i¦j, then wii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).

Connected graphs of fixed order and size with maximal Aα-index: The one-dominating-vertex case

For a simple undirected graph G, the corresponding signless Laplacian matrix is defined as D(G) + A(G) in which D(G) and A(G) are degree matrix and adjacency matrix of G, respectively. The graph G is said to be determined by its signless Laplacian spectrum, if any graph having the same signless Laplacian spectrum as G is isomorphic to G. Also the Sun graph of order 2n is a cycle Cn with an edge terminating in a pendent vertex attached to each vertex. Among other things, one result in this paper is that the Sun graphs are determined by their signless Laplacian spectrum.

Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs

Multikernel adaptive filtering over graphs based on normalized LMS algorithm

In this thesis, we apply spectral graph theory to show the non-existence of an expander family within the class of circulant graphs. Using the adjacency matrix and its properties, we prove Cheeger's inequalities and determine when the equalities hold. In order to apply Cheeger's inequalities, we compute the spectrum of a general circulant graph and approximate its second largest eigenvalue. Finally, we show that circulant graphs do not contain an expander family.

Ordering graphs with index in the interval [formula omitted

In geographic, the eigenvalues and eigenvectors of transportation network provides many informations about its connectedness. It is proven that the more highly connected in a transportation network G has largest eigenvalue and hence more multiple occurrences of the eigenvalue -1. For a graph G with adjacency matrix A, the multiplicity of the eigenvalue -1 equals the dimension of the null space of the matrix A + I. In this paper, we constructed a high closed zero sum weighting of G and by which its proved that, the dimension of the null space of the eigenvalue -1 is the same as the number of independent variables used in a non-trivial high closed zero sum weighting of the graph. Multiplicity of -1 as an eigenvalue of known graphs and of corona product of certain classes of graphs are determined and two classes of -1- nut graphs are constructed.

A method for computing local contributions to graph energy based on Estrada–Benzi approach

Fault diagnosis of roller bearings based on Laplacian energy feature extraction of path graphs

Let G be a graph and denote by L(G) = D( G) — A(G) the difference between the diagonal matrix of vertex degrees and the adjacency matrix. Denote the spectrum of L(G) by X(G). If G 1 and G 2 are disjoint graphs, then λ( G 1.G 2) majorizes λ(G 1∪ G 2), where G 1.G 2 is any coalescence of G 1 and G 2. An application to the algebraic connectivity of a retraction is given that affirms an earlier conjecture of the authors and V. S. Sunder. An important tool used in the proof of majorization involves the factorization L(G) = Q(G) Q 1(G) where Q(G) is the vertex-edge incidence matrix of G. We are able to use Q(G) to introduce edge valuations of G, particularly when G is a tree.