Spectra of Indu–Bala product of graphs and some new pairs of cospectral graphs

2019 ◽  
Vol 11 (05) ◽  
pp. 1950056
Author(s):  
Shreekant Patil ◽  
Mallikarjun Mathapati
Keyword(s):  
Spanning Trees ◽  
Product Formula ◽  
Regular Graphs ◽  
Kirchhoff Index ◽  
Cospectral Graphs ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Graph Operation ◽  
Product Of Graphs ◽  
Number Of Spanning Trees

Recently Indulal and Balakrishnan [Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graph Comb. 13 (2016) 230–234] put forward a new graph operation, namely, the Indu–Bala product [Formula: see text] of graphs [Formula: see text] and [Formula: see text], and it is obtained from two disjoint copies of the join [Formula: see text] of [Formula: see text] and [Formula: see text] by joining the corresponding vertices in the two copies of [Formula: see text]. In this paper, we obtain the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of [Formula: see text] in terms of the corresponding spectra of [Formula: see text] and [Formula: see text]. As applications, these results enable us to construct infinitely many pairs of respective cospectral graphs. Further, the Laplacian spectra enable us to get the formulas of the number of spanning trees and Kirchhoff index of [Formula: see text] in terms of the Laplacian spectra of regular graphs [Formula: see text] and [Formula: see text].

scholarly journals On the spectra of a new duplication based corona of graphs

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 ◽  
Number Of Spanning Trees

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.

scholarly journals Spectra of Subdivision Vertex-Edge Join of Three Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 171 ◽  
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 ◽  
Number Of Spanning Trees

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.

scholarly journals Generalized Characteristic Polynomials of Join Graphs and Their Applications

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 ◽  
Number Of Spanning Trees

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.

Study on Applications of Laplacian Spectra for a Network

2013 ◽  
Vol 753-755 ◽  
pp. 2859-2862
Author(s):  
Hai Tang Wang
Keyword(s):  
Recurrence Relations ◽  
Spanning Trees ◽  
Small World ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Scale Free ◽  
Structure And Dynamics ◽  
Laplacian Spectra ◽  
Number Of Spanning Trees ◽  
Biological Field

Systems composing of dynamical units are ubiquitous in nature, ranging from physical to technological, and to biological field. These systems can be naturally described by networks, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics for a network. In this paper, we study the Laplacian spectra of a family with scale-free and small-world properties. Based on the obtained recurrence relations, we determine explicitly the product of all nonzero Laplacian eigenvalues, as well as the sum of the reciprocals of these eigenvalues. Then, using these results, we further evaluate the number of spanning trees, Kirchhoff index.

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

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1374
Author(s):  
Umar Ali ◽  
Hassan Raza ◽  
Yasir Ahmed
Keyword(s):  
Structural Properties ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Molecular Graph ◽  
Decomposition Theorem ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Polynomial Decomposition ◽  
Number Of Spanning Trees

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.

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

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
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 $q(G)$).

scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

Heuristic maximization of the number of spanning trees in regular graphs

2006 ◽  
Vol 343 (3) ◽  
pp. 309-325
Author(s):  
Jacek Wojciechowski ◽  
Jarosław Arabas ◽  
Błażej Sawionek
Keyword(s):  
Spanning Trees ◽  
Regular Graphs ◽  
Number Of Spanning Trees
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