‎Let $G$ be a graph without an isolated vertex‎, ‎the normalized Laplacian matrix $tilde{mathcal{L}}(G)$‎ ‎is defined as $tilde{mathcal{L}}(G)=mathcal{D}^{-frac{1}{2}}mathcal{L}(G)mathcal{D}^{-frac{1}{2}}$‎, where ‎$mathcal{D}$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎G‎$‎‎. ‎The eigenvalues of‎ $tilde{mathcal{L}}(G)$ are ‎called as ‎the ‎normalized Laplacian eigenva...

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

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 introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=sum_{i=1}^n |eta_i|, where eta_i=zeta_i-frac{2m}{n} and where zeta_1,zeta_2,ldots,zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if sum_{i=1}^n |eta_i|=sum_{i=1}^n |zeta_i|. Conditions ...

For a graph G, let S(G) be the Seidel matrix of G and θ1(G), . . . , θn(G) be the eigenvalues of S(G). The Seidel energy of G is defined as |θ1(G)| + · · · + |θn(G)|. Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least 2n − 2, the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any α with 0 < α < 2, |θ1(...

Among all trivalent graphs on n vertices, let Gn be one with the smallest possible eigenvalue gap. (The eigenvalue gap is the difference between the two largest eigenvalues of the adjacency matrix; for regular graphs, it equals the second smallest eigenvalue of the Laplacian matrix.) We show that Gn is unique for each n and has maximum diameter. This extends work of Guiduli and solves a conject...

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...

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...

A Seidel-type interface condition is considered for non-overlapping domain decomposition iterative methods. With a suitable pseudo-energy defined on interfaces, the convergence speed of the iterative scheme is shown to be as twice fast as that of the Jacobi scheme. Our analysis is entirely independent of the governing model problems of a specific type of partial differential equations, but depe...