The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomo...

In this paper, a new virtual leader following consensus protocol is introduced to perform the internal and string stability analysis of longitudinal platoon of vehicles under generic network topology. In all previous studies on multi-agent systems with generic network topology, the control parameters are strictly dependent on eigenvalues of network matrices (adjacency or Laplacian). Since some ...

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrödinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for th...

Randić index is one of the most famous topological graph indices. The energy a was defined more than four decades ago for its molecular applications. classical modeling molecule as sum absolute values all eigenvalues adjacency matrix corresponding to graph. There are several other versions notion obtained using types matrices. In this paper, we introducing and investigating type Hadi RHE(G) G, ...

This is the third part of our work with a common title. The first [11] and the second part [12] will be also referred in the sequel as Part I and Part II, respectively. This third part was not planned at the beginning, but a lot of recently published papers on the signless Laplacian eigenvalues of graphs and some observations of ours justify its preparation. By a spectral graph theory we unders...

The Laplacian spectra are the eigenvalues of Laplacian matrix L(G) = D(G) - A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical proper...

Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ1 > λ2 > · · · > λd. For every k = 0, 1, . . . , d−1, the k-alternating polynomial Pk is defined to be the polynomial of degree k and norm ‖Pk‖∞ = max1≤l≤d{|Pk(λl)|} = 1 that attains maximum value at λ. These polynomials, which may be thought of as the discrete version of the Chebychev ones, were recently used by ...

Let (G) and min (G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) If H is a proper subgraph of a connected graph G of order n and diameter D; then (G) (H) > 1 2D (G)n : (ii) If G is a connected nonbipartite graph of order n and diameter D, then (G) + min (G) > 2 2D (G)n : These bounds have the correct order of magnitude for large and D. ...

The need to build a link between the structure of a complex network and the dynamical properties of the corresponding complex system (comprised of multiple low dimensional systems) has recently become apparent. Several attempts to tackle this problem have been made and all focus on either the controllability or synchronisability of the network — usually analyzed by way of the master stability f...

Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of q communities, the number of eigenvectors corresponding to the q largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modu...