Laplacian Polynomial Research Articles

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Study of Random Walk Invariants for Spiro-Ring Network Based on Laplacian Matrices

The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state j in a system from a starting state i. In contrast, there exists an intrinsic measure for a stochastic process, known as Kemeny’s constant, which is independent of the initial state. In the literature, it has been used as a measure of network efficiency. This article deals with a graph-spectrum-based method for finding both the GMFPT and Kemeny’s constant of random walks on spiro-ring networks (that are organic compounds with a particular graph structure). Furthermore, we calculate the Laplacian matrix for some specific spiro-ring networks using the decomposition theorem of Laplacian polynomials. Moreover, using the coefficients and roots of the resulting matrices, we establish some formulae for both GMFPT and Kemeny’s constant in these spiro-ring networks.

On the Laplacian Coefficients of Trees

Let G be a finite simple graph with Laplacian polynomial ψ(G,λ) = ∑ k=0n(−1)n−kck(G)λk. In an earlier paper, we computed the coefficient of cn−4 for trees with respect to some degree-based graph invariant. The aim of this paper is to continue this work by giving an exact formula for the coefficient cn−5 in the polynomial ψ(G,λ). As a consequence of this work, the Laplacian coefficients cn−k, k = 2, 3, 4, 5, for some know trees were computed.

On Graphs with c2-c3 Successive Minimal Laplacian Coefficients

Let G be a graph of order n and L(G) be its Laplacian matrix. The Laplacian polynomial of G is defined as P(G;λ)=det(λI−L(G))=∑i=0n(−1)ici(G)λn−i, where ci(G) is called the i-th Laplacian coefficient of G. Denoted by Gn,m the set of all (n,m)-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each i(i=0,1,…,n), H is ci(G)-minimal in Gn,m. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in Gn,m because such graphs have minimal Laplacian-like energy. We determine that the c2(G)-c3(G) successive minimal graph is exactly one of the four classes of threshold graphs.

Matrix Analysis of Hexagonal Model and Its Applications in Global Mean-First-Passage Time of Random Walks

Recent advances in graph-structured learning have demonstrated promising results on the graph classification task. However, making them scalable on huge graphs with millions of nodes and edges remains challenging due to their high temporal complexity. In this paper, by the decomposition theorem of Laplacian polynomial and characteristic polynomial we established an explicit closed-form formula of the global mean-first-passage time (GMFPT) for hexagonal model. Our method is based on the concept of GMFPT, which represents the expected values when the walk begins at the vertex. GMFPT is a crucial metric for estimating transport speed for random walks on complex networks. Through extensive matrix analysis, we show that, obtaining GMFPT via spectrums provides an easy calculation in terms of large networks.

EXISTENCE OF PERIODIC SOLUTIONS FOR TWO CLASSES OF SECOND ORDER <inline-formula><tex-math id="M1">$ P $</tex-math></inline-formula>-LAPLACIAN DIFFERENTIAL EQUATIONS

In this paper, by using the Man$ \acute{a} $sevich-Mawhin theorem on continuity of the topological degree, we prove the existence of periodic solutions for two classes of second order $ p $-Laplacian polynomial differential equations. Finally, some examples are given to show applications of the conclusions.

Consensus analysis of the weighted corona networks

The consensus of complex networks has attracted the attention of many scholars. The graph operation is a common method to construct complex networks, which is helpful in studying the consensus of complex networks. Based on the corona networks G1◦G2, this study gives different weights to the edges of G1◦G2 to obtain the weighted corona networks G̃1◦G̃2 and studies the consensus of G̃1◦G̃2. The consensus of the networks can be measured by coherence. First, the Laplacian polynomial of G̃1◦G̃2 is derived by using the properties of an orthogonal matrix. Second, the relationship between the first-order coherence of G̃1◦G̃2 and G1 is deduced by using the relevant properties of the determinant and the conclusion of polynomial coefficients and the principal minors of the matrix. Third, the join operation is introduced to further simplify the analytical formula of network coherence. Finally, a specific network example is used to verify the validity of the conclusion.

The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo <img src=image/13424080_01.gif>

Let <img src=image/13424080_02.gif> be a simple graph of order <img src=image/13424080_03.gif> and let <img src=image/13424080_04.gif> be the Seidel matrix of <img src=image/13424080_02.gif>, defined as <img src=image/13424080_05.gif> where <img src=image/13424080_06.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are adjacent and <img src=image/13424080_09.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are not adjacent and <img src=image/13424080_10.gif> if <img src=image/13424080_11.gif>. Let <img src=image/13424080_12.gif> be the diagonal matrix where <img src=image/13424080_13.gif> denotes the degree of the <img src=image/13424080_14.gif> vertex of <img src=image/13424080_02.gif>. The Seidel Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_15.gif> and the Seidel signless Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_16.gif>. The zero-divisor graph of a commutative ring <img src=image/13424080_17.gif>, denoted by <img src=image/13424080_18.gif>, is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices <img src=image/13424080_19.gif> are adjacent if and only if <img src=image/13424080_20.gif>. In this paper, we find the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler's lemma and apply these results to describe these eigenvalues for the zero-divisor graph on <img src=image/13424080_21.gif>. Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of <img src=image/13424080_21.gif> for some values of <img src=image/13424080_03.gif>, say <img src=image/13424080_22.gif>, where <img src=image/13424080_23.gif> are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of <img src=image/13424080_24.gif>, for any <img src=image/13424080_03.gif>.

The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

Spectral analysis of three invariants associated to random walks on rounded networks with 2n-pentagons

A network is defined as an abstract structure that consists of nodes that are connected by links. In this paper, we study two types of rounded networks (resp., ). By using the recursive relation, we obtain all the eigenvalues and their multiplicities with regards to the associated Laplacian matrix. Meanwhile, we utilize the corresponding relationship between the roots and the coefficients of the characteristic polynomial. Based on these relationships, we obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. By the decomposition theorem of Laplacian polynomial, we obtained an explicit closed-form formula of the Kirchhoff index for . As an application of the Laplacian spectra, we reduce the number of spanning trees and the global mean-first passage time for . Furthermore, we show that the Kirchhoff index of is approximately to of its Wiener index. In view of our obtained results, all the corresponding results are considered for .

On the Kirchhoff Index and the Number of Spanning Trees of Linear Phenylenes Chain

Let denote a molecular graph of linear phenylene, containing n hexagons and squares. In this paper, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of consists of the Laplacian spectrum of path and eigenvalues of a symmentric tridiagonal matrix of order 4n. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above matrix, explicit closed formula of Kirchhoff index and the number of spanning trees of are derived in terms of the Laplacian spectrum.

Laplacian spectra of power graphs of certain prime-power Abelian groups

The power graph [Formula: see text] of a semigroup [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] itself, and any two distinct vertices are adjacent if one of them is a power of the other. In this paper, we describe the power graph [Formula: see text] in terms of joins and disjoint unions of complete graphs, and use this to calculate the Laplacian polynomial of [Formula: see text]. We finally calculate the Laplacian polynomial of the power graph [Formula: see text].

Harary polynomials

Given a graph property $\mathcal{P}$, F. Harary introduced in 1985 $\mathcal{P}$-colorings, graph colorings where each colorclass induces a graph in $\mathcal{P}$. Let $\chi_{\mathcal{P}}(G;k)$ counts the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. It turns out that $\chi_{\mathcal{P}}(G;k)$ is a polynomial in $\mathbb{Z}[k]$ for each graph $G$. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial $\chi(G;k)$. We show that the characteristic and Laplacian polynomial, the matching, the independence and the domination polynomial are not Harary polynomials. We show that for various notions of sparse, non-trivial properties $\mathcal{P}$, the polynomial $\chi_{\mathcal{P}}(G;k)$ is, in contrast to $\chi(G;k)$, not a chromatic, and even not an edge elimination invariant. Finally we study whether Harary polynomials are definable in Monadic Second Order Logic.

On a poset of trees revisited

This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikvári (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and recurrence relation, we extend the works of Csikvári (2010) [4] and Csikvári (2013) [5]. Using a unified approach, we obtain the following main results: Firstly, for all n and ℓ, we characterize the unique tree having the maximum (resp. minimum) Wiener index and the unique tree having the maximum (resp. minimum) number of closed walks of length ℓ among the trees of order n which are neither a path nor a star. Secondly, we characterize the unique tree whose adjacency polynomial (resp. Laplacian polynomial, independence polynomial) has the maximum (resp. minimum) coefficients in absolute value among the trees of order n which are neither a path nor a star, respectively. At last, we identify all the n-vertex trees whose edge cover polynomial has the minimum coefficients in absolute value and we also determine the unique tree having the maximum coefficients in absolute value among the trees of order n which are neither a path nor a star.

Normalized Laplacian polynomial of n-Cayley graphs

Let G be a finite group and Γ be a (di)graph. Then Γ is called an n-Cayley (di)graph over G if admits a semiregular subgroup isomorphic to G with n orbits on . In this paper, we determine the normalized Laplacian polynomial of n-Cayley (di)graphs over a group G in terms of irreducible representations of G. We give exact formulas for the normalized Laplacian eigenvalues of 2-Cayley graphs over abelian groups. Among other results, as an application, we prove that the degree-Kirchhoff index of the n-sunlet graph is .

On the Kirchhoff Indices and the Numbers of Spanning Trees of Two Types of Molecular Graphs

Let Ln be the linear phenylene. Then let be the molecular graph obtained from Ln by connecting each pair of non-adjacent vertices on each 4-cycle of Ln by an edge. In this paper, according to the decomposition theorem of Laplacian polynomial, we study the Laplacian spectra of Ln and respectively. By applying the relationship between the roots and coefficients of the characteristic polynomial of Ln (resp. ), explicit closed formulae of Kirchhoff index and the number of spanning trees of Ln and are, respectively, derived in terms of the corresponding Laplacian spectrum. Furthermore, it is surprising to find that the Kirchhoff index of Ln is approximately to one half of its Wiener index, whereas the Kirchhoff index of is approximately to of its Wiener index.

The average Laplacian polynomial of a graph

Let G=(V(G),E(G)) be a graph of order n=|V(G)| and size m=|E(G)|, and let Σ(G)={σ:E(G)→{−1,+1}} be the set of sign functions defined on the edges of G. Then, it is well known that μ(G,x)=12m∑σ∈Σ(G)det(xIn−A(Gσ)), where μ(G,x) is the matching polynomial of G, and A(Gσ) is the adjacency matrix of the signed graph Gσ=(G,σ). Motivated by this result, in this paper, we introduce the average Laplacian polynomial of G, which is defined as ψ¯(G,x)=12m∑σ∈Σ(G)det(xIn−L(Gσ)), where L(Gσ) is the Laplacian matrix of the signed graph Gσ. We find that this polynomial is closely related to the structure of the graph. For example, its coefficients can be expressed naturally in terms of the TU-subgraphs of G, and the multiplicity of 0 as a root is equal to the number of tree components in G. The relations between the average Laplacian polynomial of G and other polynomials, especially, the matching polynomial, are also investigated in this paper. Based on the relations, we prove that the roots of the average Laplacian polynomial of any graph are non-negative real numbers.

On the Kirchhoff Indices and the Number of Spanning Trees of Möbius Phenylenes Chain and Cylinder Phenylenes Chain

Let denote a molecular graph of linear phenylene. Then the Möbius phenylenes chain is the graph obtained from the by identifying the opposite lateral edges in reversed way, whereas the cylinder phenylene chain is the graph obtained from the by identifying the opposite lateral edges in ordered way. In this paper, by the decomposition theorem of Laplacian polynomial and the relationship between the roots and the coefficients of the characteristic polynomial, explicit closed-form formulae of the Kirchhoff index (resp. the number of spanning trees) of and are obtained.