Graph algorithms that test adjacencies are usually implemented with an adjacency-matrix representation because the adjacency takes constant time matrices, but it linear in degree of vertices lists. In this article, we review adjacency-map representation, which supports tests expected time, and show graph run faster maps than lists by a small factor if they do not one or two orders magnitude per...

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let N be a matrix entries which have symmetric distribution. For each unit eigenvector v our main results provide small ball probability bound for linear combinations coordinates v. Our generalize works Meehan Nguyen [59] as well Touri second author [67, 68, 69] matrices. Along way, we an optimal estimate that has si...

A fundamental problem in the study of networks is identification important nodes. This typically achieved using centrality metrics, which rank nodes terms their position network. approach works well for static networks, that do not change over time, but does consider dynamics Here we propose instead to measure importance a node based on how much its strength will impact global structure network...

We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. closure, boundary, connected components, and topological properties e.g. homology groups. Other applications include visualization and path planning. Our algorithm determines cell adjacency information using...

In this paper, a general way to determine the adjacency graph of linear feedback shift registers (LFSRs) with characteristic polynomial (1 + x)c(x) from the adjacency graph of LFSR with characteristic polynomial c(x) is discussed, where c(x) can be any polynomial. As an application, the adjacency graph of LFSRs with characteristic polynomial (1+ x)p(x) are determined, where p(x) is a primitive ...

For a graph $G$, we associate family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location nonzero off-diagonal entries $M$ is governed by adjacency structure $G$. The ordered multiplicity Inverse Eigenvalue Problem Graph (IEPG) concerned with finding all attainable lists eigenvalue multiplicities matrices in $\mathcal{S}(G)$. connected graphs order si...

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with graph. In this paper, we show several NP-hard zero forcing numbers are not spectra types matrices particular, consider standard forcing, positive semidefinite and skew provide constructions infinite families pairs cospectral graphs, which different ...

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency metric dimension and the local metric dimension have been introduced. We combine these variants and introduce the local adjacency metric dimension. We show that the (local) metric dimension of the corona product of a graph of order n and some non-trivial graph H equals n times the (local) adjacency metric dime...

Consider two graphs G and H. Let H[G] be the lexicographic product of H and G, where H is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H[G] and H whenG andH are regular and the Laplacian spectrum ofH[G] andH for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic p...