In this paper, we study synchronization of complex random networks of nonlinear oscillators, with specifiable expected degree distribution. We review a sufficient condition for synchronization and a sufficient condition for desynchronization, expressed in terms of the eigenvalue distribution of the Laplacian of the graph and the coupling strength. We then provide a general way to approximate th...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

Among all trivalent graphs on n vertices, let Gn be one with the smallest possible eigenvalue gap. (The eigenvalue gap is the difference between the two largest eigenvalues of the adjacency matrix; for regular graphs, it equals the second smallest eigenvalue of the Laplacian matrix.) We show that Gn is unique for each n and has maximum diameter. This extends work of Guiduli and solves a conject...

Given the topology of a graph G and a budget k, how can we quickly find the best k edges to delete that minimize dissemination in G? Stopping dissemination in a graph is important in a variety of fields from epidemiology to cyber security. The spread of an entity (e.g., a virus) on an arbitrary graph depends on two properties: (1) the topology of the graph and (2) the characteristics of the ent...

Let $$\Gamma =(K_{n},H^-)$$ be a signed complete graph whose negative edges induce subgraph H. The index of $$\Gamma$$ is the largest eigenvalue its adjacency matrix. In this paper, we study when H unicyclic graph. We show that among all graphs order $$n>5$$ k and maximizes index, triangle with remaining vertices being pendant at same vertex triangle.

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros of the nth eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the nth eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the nth eigenvalue as a function of the magnetic perturbation and show that its Morse index at zer...

We prove that the minimum value of the least eigenvalue of the signless Laplacian of a connected nonbipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices. © 2008 Elsevier Inc. All rights reserved. AMS classification: 05C50

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively.

We consider the general problem of determining the maximum possible multiplicity of an eigenvalue in a Hermitian matrix whose graph contains exactly one cycle. For some cases we express that maximum multiplicity in terms of certain parameters associated with the graph. © 2006 Elsevier Inc. All rights reserved. AMS classification: 15A18; 05C38; 05C50

We give a strongly polynomial time combinatorial algorithm to minimise the largest eigenvalue of the weighted Laplacian of a bipartite graph G = (W ∪B,E). This is accomplished by solving the dual graph embedding problem which arises from a semidefinite programming formulation. In particular, the problem for trees can be solved in time O(|W ∪B|).