The universal adjacency matrix U of a graph Γ, with A, is linear combination the diagonal D vertex degrees, identity I, and all-1 J real coefficients, that is, U=c1A+c2D+c3I+c4J, ci∈R c1≠0. Thus, in particular cases, may be matrix, Laplacian, signless Seidel matrix. In this paper, we develop method for determining spectra bases all corresponding eigenspaces arbitrary lifts graphs (regular or no...

We analyze heterogeneous site percolation and the corresponding uniqueness transition on directed and undirected graphs, by constructing upper bounds on the in-/out-cluster susceptibilities and the vertex connectivity function. We give separate bounds on finite and infinite (di)graphs, and analyze the convergence in the infinite graph limit. We also discuss the transition associated with prolif...

It is known that the problem of computing adjacency dimension a graph NP-hard. This suggests finding for special classes graphs or obtaining good bounds on this invariant. In work we obtain general G in terms parameters . We discuss tightness these and, some particular graphs, closed formulae. particular, show close relationships exist between and other parameters, like domination number, locat...

Felsner, Li and Trotter showed that the dimension of adjacency poset an outerplanar graph is at most 5, gave example whose has 4. We improve their upper bound to 4, which then best possible.

The Moore bound M(k, g) is a lower bound on the order of k-regular graphs of girth g (denoted (k, g)-graphs). The excess e of a (k, g)-graph of order n is the difference n −M(k, g). In this paper we consider the existence of (k, g)-bipartite graphs of excess 4 by studying spectral properties of their adjacency matrices. For a given graph G and for the integers i with 0 6 i 6 diam(G), the i-dist...

We prove that the bulk eigenvectors of sparse random matrices, i.e. the adjacency matrices of ErdősRényi graphs or random regular graphs, are asymptotically jointly normal, provided the averaged degree increases with the size of the graphs. Our methodology follows [6] by analyzing the eigenvector flow under Dyson Brownian motion, combining with an isotropic local law for Green’s function. As an...

I study the U.S. Congressional voting record using network theory and computations. I encode the roll call votes from both the House of Representatives and the Senate for 1789-2006 into adjacency matrices (graphs) that encode the extent of agreement between any two legislators in a given House or Senate. I apply concepts such as modularity and various notions of centrality to the adjacency matr...

Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic iftheir skew energies...