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...

The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let C and K w denote the Clebsch graph and a complete graph on w vertices, respectively. In this paper, we show that the multicone graphs K w ▽C are determined by their signless ...

‎by considering a degenerate $p(x)-$laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent sobolev space is presented‎. ‎multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding nehari manifold‎.

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/...

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 ...

Eigenvalues of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called its spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, perfect matching etc. Laplacian spectrum is the multiset of eigenvalues of Laplacian matrix. The Laplacian energy of a graph is the sum of the...

A Laplacian matrix, L = (lij) ∈ R , has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with − 1 n ≤ lij ≤ 0 at j 6= i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrice...

A signless Laplacian eigenvalue of a graph G is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, some necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues are given. And the trees and unicyclic graphs with exactly two main signless L...

‎By considering a degenerate $p(x)-$Laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent Sobolev space is presented‎. ‎Multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding Nehari manifold‎. 

This work concerns the study of existence and uniqueness to heat equation with fractional Laplacian dierentiation in extended Colombeau algebra.