The Optimal Graph Whose Least Eigenvalue is Minimal among All Graphs via 1-2 Adjacency Matrix

All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 , i f v i = v j o r d v i , v j ≥ 2 1 , i f d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 , i f v i = v j o r d v i , v j ≥ 3 1 , i f d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .

Optical flow estimation using the Fisher–Rao metric

The optical flow in an event camera is estimated using measurements in the address event representation (AER). Each measurement consists of a pixel address and the time at which a change in the pixel value equalled a given fixed threshold. The measurements in a small region of the pixel array and within a given window in time are approximated by a probability distribution defined on a finite set. The distributions obtained in this way form a three dimensional family parameterized by the pixel addresses and by time. Each parameter value has an associated Fisher–Rao matrix obtained from the Fisher–Rao metric for the parameterized family of distributions. The optical flow vector at a given pixel and at a given time is obtained from the eigenvector of the associated Fisher–Rao matrix with the least eigenvalue. The Fisher–Rao algorithm for estimating optical flow is tested on eight datasets, of which six have ground truth optical flow. It is shown that the Fisher–Rao algorithm performs well in comparison with two state of the art algorithms for estimating optical flow from AER measurements.

Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions

We study the Stokes eigenvalue problem under Navier boundary conditions in $$C^{1,1}$$ C 1 , 1 -domains $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 . Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments.

The Least Eigenvalue of the Complement of the Square Power Graph of G

Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .

Line graphs of complex unit gain graphs with least eigenvalue -2

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

Least eigenvalue of the connected graphs whose complements are cacti

Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each Cc ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, where $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.