One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix M , defined in terms of the adjacency matrix and a rank one null m...

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the e...

Hoffman proved that for a simple graph G, the chromatic number χ(G) obeys χ(G) ≥ 1 − λ1 λn where λ1 and λn are the maximal and minimal eigenvalues of the adjacency matrix of G respectively. Lovász later showed that χ(G) ≥ 1− λ1 λn for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lovász’s theorem, then extend the technique to derive generali...

Community structure analysis is a powerful tool for complex networks, which can simplify their functional analysis considerably. Many approaches have recently been proposed to the communities in complex networks, but a method to characterize the node importance to communities is still lacking. In this paper a centrality metric is proposed to measure the importance of network nodes to community ...

Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction these objects, and the properties of their adjacency Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making closely related notions strongly regular graphs. also critical groups Adinkras, particular determine odd components. A novel technique indep...

For a simple connected graph G of order n, the normalized Laplacian is square matrix defined as [Formula: see text], where text] diagonal whose i-th entry text]. In this paper, we find eigenvalues joined union regular graphs in terms adjacency and quotient associated with G. finite group power which two distinct vertices are by an edge if only one other. As consequence graphs, investigate cyclic

Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...

Given a simple, undirected graph G, Budinich [Discrete Applied Mathematics, 127 (2003), 535 – 543] proposed a lower bound on the clique number of G by combining the quadratic programming formulation of the clique number due to Motzkin and Straus (1965) with the spectral decomposition of the adjacency matrix of G. This lower bound improves the previously known spectral lower bounds on the clique...

Write (A) = 1 (A) min (A) for the eigenvalues of a Hermitian matrix A. Our main result is: let A be a Hermitian matrix partitioned into r r blocks so that all diagonal blocks are zero. Then for every real diagonal matrix B of the same size as A; (B A) B + 1 r 1 : Let G be a nonempty graph, (G) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality impli...

In this paper, the design of highly synchronizable, sparse and robust dynamical networks is addressed. Better synchronizability means faster synchronization of the oscillators, sparsity means a low ratio of links per nodes and robustness refers to the resilience of a network to the random failures or intentional removal of some of the nodes/links. Golden spectral dynamical networks (graphs) are...