The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze ext...

Let A be a self-adjoint operator acting over a space X endowed with a partition. We give lower bounds on the energy of a mixed state ρ from its distribution in the partition and the spectral density of A. These bounds improve with the refinement of the partition, and generalize inequalities by Li-Yau and Lieb–Thirring for the Laplacian in R. They imply an uncertainty principle, giving a lower b...

The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the num...

Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian G,where D(G) are degree diagonal adjacency matrix, respectively. vertex sequence d1 ≥ d2 ≥··· dn let μ1 μ2 μ n−1 &gt;μn = 0 eigenvalues G. invariants, energy (LE), Laplacian-energy-like invariant (LEL) Kirchhoff index (Kf), defined in terms G, as <m:math xmlns:m="http://www.w3.o...

In this paper, some inequality relations between the Laplacian/signless Laplacian H-eigenvalues and the clique/coclique numbers of uniform hypergraphs are presented. For a connected uniform hypergraph, some tight lower bounds on the largest Laplacian H+-eigenvalue and signless Laplacian H-eigenvalue related to the clique/coclique numbers are given. And some upper and lower bounds on the clique/...

The positive-definiteness of the Hamiltonian operator for Yang–Mills theory in four dimensions is studied by using the Coulomb gauge. It was indeed well known that the Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P built from infinitely many inverse powers of the Laplacian. If space-time is replaced by a compact Riemannian four-manifold without boundary, on w...

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

Let G be a connected graph of order n with Laplacian eigenvalues [Formula: see text]. The Laplacian-energy-like invariant of G, is defined as [Formula: see text]. In this paper, we investigate the asymptotic behavior of the 3.6.24 lattice in terms of Laplacian-energy-like invariant as m, n approach infinity. Additionally, we derive that [Formula: see text], [Formula: see text] and [Formula: see...