Laplacian Characteristic Polynomial Research Articles

Deletion-contraction for a unified Laplacian and applications

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees.

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

New constructions of nonregular cospectral graphs

Abstract We consider two types of joins of graphs G 1 {G}_{1} and G 2 {G}_{2} , G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} – the neighbors splitting join and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} – the nonneighbors splitting join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial, and the signless Laplacian characteristic polynomial of these joins. When G 1 {G}_{1} and G 2 {G}_{2} are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} , and the normalized Laplacian spectrum of G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} . We use these results to construct nonregular, nonisomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian, signless Laplacian and normalized Laplacian.

Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let Pn(Pn′) represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of Pn contain the eigenvalues of matrices LR and LS. Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of Pn are determined. The relationship between the Wiener and Kirchhoff indices of Pn is also established.

On the Laplacian spectra and the Kirchhoff indices of two types of networks

Let G be a connected simple graph. Denoted by the Kirchhoff index of G. In this note, we study the Laplacian spectra of two types of networks and . As a special case, explicit formulas for the Kirchhoff index and number of spanning trees of prisms are obtained. Some other interesting results, such as the expression for the coefficient of the Laplacian characteristic polynomials of cycles, are also derived.

On (distance) Laplacian characteristic polynomials of power graphs

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are primes, are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

Combinatorial explanation of the weighted Laplacian characteristic polynomial of a graph and applications

Assume that G is a graph with n vertices v1,v2,…,vn. Given a rooted spanning forest F of G, which has s components and roots vk1,vk2,…,vks, define the weight ω(F) of F to be ∏i=1sxki, where x1,x2,…,xn are n independent commuting variables. In this paper, we give combinatorial explanations of the weighted enumeration of rooted spanning forests of G, the Laplacian characteristic polynomial of G, and the weighted Laplacian det[L+diag(x1,x2,…,xn)], where L is the Laplacian matrix of G and diag(x1,x2,…,xn) is a diagonal matrix. As applications, we count rooted spanning forests in the almost complete bipartite graph, and we also give a new proof of a formula on the number of rooted spanning forests in a complete multipartite graph Ka1,a2,…,as each of which has bi roots in the i-th part for i=1,2,…,s.

On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L(D)=Deg+(D)−A(D) and Q(D)=Deg+(D)+A(D), where A(D) represents the adjacency matrix and Deg+(D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations.

Kirchhoff index and complexity of linear Möbius and cylinder octagonal‐quadrilateral networks with respect to Laplacians

AbstractLet represent a linear octagonal‐quadrilateral network, consisting of n eight‐member rings and n four‐member rings. Such a graph contains a unique pair of opposite edges. The Möbius graph Qn(8, 4) is constructed by reverse identifying these opposite edges, whereas the cylinder graph Qn′(8, 4) identifies the opposite edges in the natural manner. In this paper, the explicit formulas for the Kirchhoff index and complexity of Qn(8, 4) and Qn′(8, 4) are deduced from Laplacian characteristic polynomials using to decomposition theorem and Vieta's theorem. A consequence is the surprising fact that the Kirchhoff index of Qn(8, 4) (resp. Qn′(8, 4)) is approximately a third (resp. half) of its Wiener index as .

On the coefficients of skew Laplacian characteristic polynomial of digraphs

Let [Formula: see text] be a connected graph with [Formula: see text] vertices and [Formula: see text] edges. Let [Formula: see text] be the digraph obtained by orienting the edges of [Formula: see text] arbitrarily. The digraph [Formula: see text] is called an orientation of [Formula: see text] or oriented graph corresponding to [Formula: see text]. The skew Laplacian matrix of the digraph [Formula: see text] is denoted by [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] is the skew matrix and [Formula: see text] is the diagonal matrix with [Formula: see text]th diagonal entry [Formula: see text]. In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of [Formula: see text]. We provide examples of orientations of some well-known graphs to highlight the importance of our results. We conclude the paper with some observations about the skew Laplacian spectral determinations of the directed path and directed cycle.

Computing and analyzing the normalized Laplacian spectrum and spanning tree of the strong prism of the dicyclobutadieno derivative of linear phenylenes

AbstractThe study of topological indices of some special chemical graphs has become a hot topic in chemical theory. Let Ln be the dicyclobutadieno derivative of linear phenylene which consists of n hexagons and (n + 1) quadrangles and let be the strong prism of the dicyclobutadieno derivative of linear phenylenes. Firstly, we obtain that the normalized Laplacian spectrum of consists of the eigenvalues of symmetric matrices and according to the decomposition theorem of normalized Laplacian characteristic polynomial. Then, by taking full advantage of the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we determine the explicit expressions of the multiplicative degree‐Kirchhoff index and the number of spanning trees of the graph .

Chain graph sequences and Laplacian spectra of chain graphs

A graph is called a chain graph if it is bipartite and the neighbourhoods of the vertices in each colour class form a chain with respect to inclusion. In this paper, we study the relation between the degree sequences and the Laplacian spectra of chain graphs. We provide explicit formulae for the Laplacian characteristic polynomial of a chain graph, and certain properties of the Laplacian spectrum that can be deduced from its degree sequence.

On the Laplacians for Strong Product Graphs Based on Polyacene Graphs

Let Hn be the molecular graph of a linear polyacene chain. In this paper, we introduce a kind of linear polyacene cylinder graph which is the strong product of two automorphism polyacene graphs Hn and Next, we obtain that the Laplacian spectrum of consists of the eigenvalues of symmetric matrices LA and LS of order according to the decomposition theorem of Laplacian characteristic polynomial. Furthermore, by applying the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we study the Kirchhoff index and the number of spanning trees, and derive their explicit expressions in terms of n for the graph

On the Multiplicative Degree-Kirchhoff Indices and the Number of Spanning Trees of Linear Phenylene Chains

Let denote a molecular graph of linear phenylene, containing n hexagons and squares. In this paper, using the decomposition theorem of the normalized Laplacian characteristic polynomial, we obtain that the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 4n is determined. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above two matrices, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. the number of spanning trees) of is derived.

Each (n,m)-graph having the i-th minimal Laplacian coefficient is a threshold graph

Let G be a simple graph with n vertices and m edges (i.e. an (n,m)-graph) and L(G) be the Laplacian matrix of G. The Laplacian characteristic polynomial of G is defined as P(G;λ)=det⁡(λI−L(G))=∑i=0n(−1)ici(G)λn−i, where ci(G) is referred as the i-th Laplacian coefficient of G. Denote Gn,m by the set of all connected (n,m)-graphs. A connected graph H∈Gn,m is called ci-minimal if ci(H)≤ci(G) holds for each G∈Gn,m and is called uniformly minimal if H is ci-minimal for i=0,1,…,n. In this paper, we prove that each ci-minimal graph in Gn,m is a threshold graph for 2≤i≤n−2. Moreover, we prove that there does not exist uniformly minimal graphs in Gn,n+3, n≥6.

On the spectra of generalized Fibonomial and Jacobsthal-binomial graphs

In this work, we first give a more general form of the binomial, Fibonomial, and balance-binomial graphs that is called generalized Fibonomial graph. We also argue the spectra of generalized Fibonomial graph. Next, we introduce a new type of graph on Jacobsthal numbers that is called Jacobsthal-binomial graph and denoted by $JB_{n}$. We obtain the adjacency, Laplacian and signless Laplacian characteristic polynomials of $JB_{n}$, respectively. We lastly give inequalities for the adjacency, Laplacian and signless Laplacian energies of $JB_{n}$.

On the Boundary of Incidence Energy and Its Extremum Structure of Tricycle Graphs

With the wide application of graph theory in circuit layout, signal flow chart and power system, more and more attention has been paid to the network topology analysis method of graph theory. In this paper, we construct a graph transformation which can reflect the monotonicity of coefficients and reduce the number of graphs. A sharp lower bound for incidence energy in the tricyclic graphs is given and all the extremal structures are characterized. The most interesting things that we find two different classes tricyclic graphs have the same signless Laplacian characteristic polynomials and one of the extremal graphs beyond all expectations

Graphs that are cospectral for the distance Laplacian

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $\mathcal{D}^L$-cospectrality, including examples of $\mathcal{D}^L$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$, where $\delta^L_{k}$ is the coefficient of $x^k$.

Kirchhoff Indices and Numbers of Spanning Trees of Molecular Graphs Derived from Linear Crossed Polyomino Chain

Polyomino systems are widely studied in organic chemistry, especially in polycyclic aromatic compounds. Let Qn be a linear crossed polyomino chain with n complete quadrangles. In this article, we construct a family of molecular graphs obtained by randomly deleting some specified edges of Qn . By using the decomposition theorem of the Laplacian matrix characteristic polynomial, explicit formulas of Kirchhoff indices and numbers of spanning trees of those molecular graphs derived from Qn are obtained. Moreover, their Kirchhoff indices are shown to be almost one quarter of their Wiener indices.

Study on the normalized Laplacian of a penta‐graphene with applications

AbstractLet L n denote a linear pentagonal chain with 2n pentagons. The penta‐graphene (penta‐C), denoted by R n is the graph obtained from L n by identifying the opposite lateral edges in an ordered way, whereas the pentagonal Möbius ring is the graph obtained from the L n by identifying the opposite lateral edges in a reversed way. In this paper, through the decomposition theorem of the normalized Laplacian characteristic polynomial and the relationship between its roots and the coefficients, an explicit closed‐form formula of the multiplicative degree‐Kirchhoff index (resp. Kemeny's constant, the number of spanning trees) of R n is obtained. Furthermore, it is interesting to see that the multiplicative degree‐Kirchhoff index of R n is approximately of its Gutman index. Based on our obtained results, all the corresponding results are obtained for .