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In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, ..., n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs. [ABSTRACT FROM AUTHOR]

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds. [ABSTRACT FROM AUTHOR]

A fundamental organizing principle of real-world complex networked systems is modularity, where networks have interactions at different levels. In this paper we consider a modular graph G having modules with arbitrary intraconnections and random interconnections between activated vertices in different modules. The vertices in different modules are activated with probability r and linked by an interconnecting edge with probability p independently. We present results regarding the Cheeger constant, robustness, algebraic connectivity as well as the smallest eigenvalue for the Dirichlet Laplacian matrix of G with high probability. Our results suggest that r = (ln n) / n is a potential scaling for the recently observed external field-like phenomena of modular networks in statistical mechanics. [ABSTRACT FROM AUTHOR]

The reciprocal distance Laplacian matrix of a connected graph G is defined as R D L (G) = R T (G) − R D (G) , where R T (G) is the diagonal matrix of reciprocal distance degrees and R D (G) is the Harary matrix. Clearly, R D L (G) is a real symmetric matrix, and we denote its eigenvalues as λ 1 (R D L (G)) ≥ λ 2 (R D L (G)) ≥ ... ≥ λ n (R D L (G)) . The largest eigenvalue λ 1 (R D L (G)) of R D L (G) , denoted by λ (G) , is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree R T m a x , minimum reciprocal distance degree R T m i n , and Harary index H (G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ (G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread R D L S (G) = λ 1 (R D L (G)) − λ n − 1 (R D L (G)) in terms of various graph parameters. We determine the extremal graphs in many cases. [ABSTRACT FROM AUTHOR]

Given a simple graph G, the distance signlesss Laplacian DQ(G) = Tr(G) + D(G) is the sum of vertex transmissions matrix Tr(G) and distance matrix D(G). In this paper, thanks to the symmetry of DQ(G), we obtain novel sharp bounds on the distance signless Laplacian eigenvalues of G, and in particular the distance signless Laplacian spectral radius. The bounds are expressed through graph diameter, vertex covering number, edge covering number, clique number, independence number, domination number as well as extremal transmission degrees. The graphs achieving the corresponding bounds are delineated. In addition, we investigate the distance signless Laplacian spectrum induced by Indu-Bala product, Cartesian product as well as extended double cover graph. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph on n vertices. The distance Estrada index DEE(G) of G is defined as the sum of eλi(D) over 1≤i≤n, where λ1(D),λ2(D),…,λn(D) are the eigenvalues of its distance matrix D. In this paper, we establish lower and upper bounds to DEE(G) for almost all bipartite graphs G. [ABSTRACT FROM AUTHOR]

This paper deals with finite-time scaled consensus problems over undirected and directed topologies, wherein agents’ states reach prescribed ratios in a finite time. We develop distributed linear iterations as a function of a linear operator on the underlying network and present necessary and sufficient conditions guaranteeing scaled consensus in a fixed number of steps equal to the number of distinct eigenvalues of a related linear operator. We identify the dependence of the final consensus states on the initial state condition, which can be conveniently and freely tuned by designing suitable parameters. Our results extend the recently developed approach on successive nulling of eigenvalues from complete consensus to scaled consensus, and from undirected topologies to directed topologies. Numerical examples and comparison studies are provided to illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

Supposeis a simple graph andare the eigenvalues of its distance matrix. The distance Estrada indexofis defined as the sum of,. In this paper, we establish better lower and upper bounds tofor almost all graphs. [ABSTRACT FROM AUTHOR]

Let $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided. [ABSTRACT FROM AUTHOR]

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as R Q (G) = diag (R H (G)) + R D (G) . Here, R D (G) is the Harary matrix (also called reciprocal distance matrix) while diag (R H (G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph K n and the graph K n − e obtained from K n by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue. [ABSTRACT FROM AUTHOR]

Given a commutative ring R with identity 1 ≠ 0 , let the set Z (R) denote the set of zero-divisors and let Z * (R) = Z (R) \ { 0 } be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ (R) , is a simple graph whose vertex set is Z * (R) and each pair of vertices in Z * (R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ (Z n) for n = p N 1 q N 2 , where p < q are primes and N 1 , N 2 are positive integers. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by E E (G) = ∑ i = 1 n e λ i (A (G)) and L E E (G) = ∑ i = 1 n e λ i (L (G)) , where { λ i (A (G)) } i = 1 n and { λ i (L (G)) } i = 1 n are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D (G) be the distance matrix, D L (G) be the distance Laplacian, D Q (G) be the distance signless Laplacian, and T r (G) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α (G) the generalized distance matrix, i.e., D α (G) = α T r (G) + (1 − α) D (G) , where α ∈ [ 0 , 1 ] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue. [ABSTRACT FROM AUTHOR]

Suppose that G is a simple undirected connected graph. Denote by D (G) the distance matrix of G and by T r (G) the diagonal matrix of the vertex transmissions in G, and let α ∈ [ 0 , 1 ] . The generalized distance matrix D α (G) is defined as D α (G) = α T r (G) + (1 − α) D (G) , where 0 ≤ α ≤ 1 . If ∂ 1 ≥ ∂ 2 ≥ ... ≥ ∂ n are the eigenvalues of D α (G) ; we define the generalized distance Estrada index of the graph G as D α E (G) = ∑ i = 1 n e ∂ i − 2 α W (G) n , where W (G) denotes for the Wiener index of G. It is clear from the definition that D 0 E (G) = D E E (G) and 2 D 1 2 E (G) = D Q E E (G) , where D E E (G) denotes the distance Estrada index of G and D Q E E (G) denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for D α E (G) of some special classes of graphs. [ABSTRACT FROM AUTHOR]

Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by λ 1 , λ 2 , ⋯ , λ n . The Gaussian Estrada index, denoted by H (G) (Estrada et al., Chaos 27(2017) 023109), can be defined as H (G) = ∑ i = 1 n e − λ i 2 . Gaussian Estrada index underlines the eigenvalues close to zero, which plays an important role in chemistry reactions, such as molecular stability and molecular magnetic properties. In a network of particles governed by quantum mechanics, this graph-theoretic index is known to account for the information encoded in the eigenvalues of the Hamiltonian near zero by folding the graph spectrum. In this paper, we establish some new lower bounds for H (G) in terms of the number of vertices, the number of edges, as well as the first Zagreb index. [ABSTRACT FROM AUTHOR]

• This paper studies the distance Laplacian spectral ordering of sun type graphs. • The distance Laplacian eigenvalues and energies of sun and partial sun graphs are obtained. • The partial sun graphs can be ordered based on their second-largest distance Laplacian eigenvalue. • Sun type graphs can be ordered based on their distance Laplacian energies. Let G be a simple, connected graph of order n. Its distance Laplacian energy D L E (G) is given by D L E (G) = ∑ i = 1 n | ρ i L − 2 W (G) n | , where ρ 1 L ≥ ρ 2 L ≥ ⋯ ≥ ρ n L are the distance Laplacian eigenvalues and W (G) is the Wiener index of G. Distance Laplacian eigenvalues of sun and partial sun graphs have been characterized. We order the partial sun graphs by using their second largest distance Laplacian eigenvalue. Moreover, the distance Laplacian energy of sun and partial sun graphs have been derived in this paper. These graphs are also ordered by using their distance Laplacian energies. [ABSTRACT FROM AUTHOR]