Abstract. Given an undirected graph G, the classical Cheeger constant, hG, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The wellknown Cheeger’s inequality states that 2λ1 ≤ hG ≤ √ 2λ1 where λ1 is the minimum nontrivial eigenvalue of the normalized Laplacian matrix. Recent work has generalized the concept of...

Abstract The distance matrix

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 ...

In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices and present an alternative proof for finding eigenvalues of the adjacency matrix of paths and cycles using Chebyshev polynomials. Even though each results i...

The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

In this paper graphs with the largest Laplacian eigenvalue at most 4 are characterized. Using this we show that the graphs with the largest Laplacian eigenvalue less than 4 are determined by their Laplacian spectra. Moreover, we prove that ones with no isolated vertex are determined by their adjacency spectra.

For a graph G, the unraveled ball of radius r centered at vertex v is in universal cover G. We obtain lower bound on weighted spectral balls fixed with positive weights edges, which used to present an upper sth (where s≥2) smallest normalized Laplacian eigenvalue irregular graphs under minor assumptions. Moreover, when s=2, result may be regarded as Alon–Boppana type for class graphs.

We study structure, eigenvalue spectra, and random-walk dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adj...

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph...