Properties of Commuting Graphs over Semidihedral Groups

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972715000027 ◽

2015 ◽

Vol 91 (3) ◽

pp. 353-367 ◽

Cited By ~ 38

Author(s):

JING HUANG ◽

SHUCHAO LI

Keyword(s):

Lower Bound ◽

Characteristic Polynomial ◽

Regular Graph ◽

Spanning Trees ◽

Line Graph ◽

Laplacian Spectrum ◽

Kirchhoff Index ◽

Subdivision Graph ◽

Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

Calculating the normalized Laplacian spectrum and the number of spanning trees of linear pentagonal chains

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2018.05.045 ◽

2018 ◽

Vol 344 ◽

pp. 381-393 ◽

Cited By ~ 12

Author(s):

Chunlin He ◽

Shuchao Li ◽

Wenjun Luo ◽

Liqun Sun

Keyword(s):

Spanning Trees ◽

Laplacian Spectrum ◽

Some Chemistry Indices of Clique-Inserted Graph of a Strongly Regular Graph

Complexity ◽

10.1155/2021/7671212 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Chun-Li Kan ◽

Ying-Ying Tan ◽

Jia-Bao Liu ◽

Bao-Hua Xing

Keyword(s):

Regular Graph ◽

Spanning Trees ◽

Strongly Regular Graph ◽

Laplacian Spectrum ◽

Signless Laplacian Spectrum ◽

Triangular Graph ◽

Signless Laplacian ◽

Strongly Regular ◽

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph T t , which is a strongly regular graph, is enumerated.

Energy, Laplacian energy of double graphs and new families of equienergetic graphs

Acta Universitatis Sapientiae Informatica ◽

10.2478/ausi-2014-0020 ◽

2014 ◽

Vol 6 (1) ◽

pp. 89-116 ◽

Cited By ~ 8

Author(s):

Hilal A. Ganie ◽

Shariefuddin Pirzada ◽

Antal Iványi

Keyword(s):

Bipartite Graph ◽

Spanning Trees ◽

Double Cover ◽

Laplacian Energy ◽

Vertex Set ◽

Equienergetic Graphs

Abstract For a graph G with vertex set V(G) = {v1, v2, . . . , vn}, the extended double cover G* is a bipartite graph with bipartition (X, Y), X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}, where two vertices xi and yj are adjacent if and only if i = j or vi adjacent to vj in G. The double graph D[G] of G is a graph obtained by taking two copies of G and joining each vertex in one copy with the neighbors of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs G* and D[G], L-spectra of Gk* the k-th iterated extended double cover of G. We obtain a formula for the number of spanning trees of G*. We also obtain some new families of equienergetic and L-equienergetic graphs.

Generalized Characteristic Polynomials of Join Graphs and Their Applications

Discrete Dynamics in Nature and Society ◽

10.1155/2017/2372931 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10

Author(s):

Pengli Lu ◽

Ke Gao ◽

Yang Yang

Keyword(s):

Characteristic Polynomial ◽

Regular Graph ◽

Spanning Trees ◽

Characteristic Polynomials ◽

Electrical Networks ◽

Kirchhoff Index ◽

Laplacian Spectra ◽

Laplacian Energy ◽

Signless Laplacian ◽

The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices ofGin electrical networks.LEL(G)is the Laplacian-Energy-Like Invariant ofGin chemistry. In this paper, we define two classes of join graphs: the subdivision-vertex-vertex joinG1⊚G2and the subdivision-edge-edge joinG1⊝G2. We determine the generalized characteristic polynomial of them. We deduce the adjacency (Laplacian and signless Laplacian, resp.) characteristic polynomials ofG1⊚G2andG1⊝G2whenG1isr1-regular graph andG2isr2-regular graph. As applications, the Laplacian spectra enable us to get the formulas of the number of spanning trees, Kirchhoff index, andLELofG1⊚G2andG1⊝G2in terms of the Laplacian spectra ofG1andG2.

Spectra of Subdivision Vertex-Edge Join of Three Graphs

Mathematics ◽

10.3390/math7020171 ◽

2019 ◽

Vol 7 (2) ◽

pp. 171 ◽

Cited By ~ 4

Author(s):

Fei Wen ◽

You Zhang ◽

Muchun Li

Keyword(s):

Spanning Trees ◽

Regular Graphs ◽

Laplacian Spectrum ◽

Kirchhoff Index ◽

Signless Laplacian Spectrum ◽

Signless Laplacian ◽

Adjacency Spectrum ◽

Graph Operation ◽

Arbitrary Graphs ◽

In this paper, we introduce a new graph operation called subdivision vertex-edge join (denoted by G 1 S ▹ ( G 2 V ∪ G 3 E ) for short), and then the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) are respectively determined in terms of the corresponding spectra for a regular graph G 1 and two arbitrary graphs G 2 and G 3 . All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, Bull. Malays. Math. Sci. Soc., 2017:1–17]. Furthermore, we also determine the normalized Laplacian spectrum of G 1 S ▹ ( G 2 V ∪ G 3 E ) whenever G i are regular graphs for each index i = 1 , 2 , 3 . As applications, we construct infinitely many pairs of A-cospectral mates, L-cospectral mates, Q-cospectral mates and L -cospectral mates. Finally, we give the number of spanning trees, the (degree-)Kirchhoff index and the Kemeny’s constant of G 1 S ▹ ( G 2 V ∪ G 3 E ) , respectively.

On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

Mathematics ◽

10.3390/math7040314 ◽

2019 ◽

Vol 7 (4) ◽

pp. 314 ◽

Cited By ~ 2

Author(s):

Jia-Bao Liu ◽

Jing Zhao ◽

Zhongxun Zhu ◽

Jinde Cao

Keyword(s):

Spanning Trees ◽

Decomposition Theorem ◽

Laplacian Spectrum ◽

Explicit Formulas ◽

Kirchhoff Index ◽

Regular Networks ◽

Structure Properties

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n .

On the spectra of a new duplication based corona of graphs

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.1.208-220 ◽

2021 ◽

Vol 27 (1) ◽

pp. 208-220

Author(s):

Renny P. Varghese ◽

◽

D. Susha ◽

Keyword(s):

Spanning Trees ◽

Graph Product ◽

Kirchhoff Index ◽

Cospectral Graphs ◽

Laplacian Spectra ◽

Laplacian Energy ◽

Signless Laplacian ◽

Product Of Graphs ◽

In this paper we introduce a new corona-type product of graphs namely duplication corresponding corona. Here we mainly determine the adjacency, Laplacian and signless Laplacian spectra of the new graph product. In addition to that, we find out the incidence energy, the number of spanning trees, Kirchhoff index and Laplacian-energy-like invariant of the new graph. Also we discuss some new classes of cospectral graphs.

Estimating the Laplacian Energy-Like Molecular Structure Descriptor

Zeitschrift für Naturforschung A ◽

10.5560/zna.2012-0027 ◽

2012 ◽

Vol 67 (6-7) ◽

pp. 403-406 ◽

Cited By ~ 13

Author(s):

Ivan Gutman

Keyword(s):

Molecular Structure ◽

Spanning Trees ◽

Molecular Graph ◽

Upper Bounds ◽

Lower And Upper Bounds ◽

Laplacian Energy ◽

Structure Descriptor ◽

Better Than

Lower and upper bounds for the Laplacian energy-like (LEL) molecular structure descriptor are obtained, better than those previously known. These bonds are in terms of number of vertices and edges of the underlying molecular graph and of graph complexity (number of spanning trees)

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

Complexity ◽

10.1155/2022/5584167 ◽

2022 ◽

Vol 2022 ◽

pp. 1-10

Author(s):

Jia-Bao Liu ◽

Jing Chen ◽

Jing Zhao ◽

Shaohui Wang

Keyword(s):

Spanning Trees ◽

Characteristic Polynomials ◽

Laplacian Spectrum ◽

Kirchhoff Index ◽

Structure Properties

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 .