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

Complexity ◽  
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 ◽  
Number Of Spanning Trees

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.

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.

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 $q(G)$).

On the signless Laplacian spectral determination of the join of regular graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450050
Author(s):  
Lizhen Xu ◽  
Changxiang He
Keyword(s):  
Regular Graph ◽  
Spectral Determination ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

Let G be an r-regular graph with order n, and G ∨ H be the graph obtained by joining each vertex of G to each vertex of H. In this paper, we prove that G ∨ K2is determined by its signless Laplacian spectrum for r = 1, n - 2. For r = n - 3, we show that G ∨ K2is determined by its signless Laplacian spectrum if and only if the complement of G has no triangles.

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 $i,j\in V(G)$.

scholarly journals Upper bounds on Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

2017 ◽  
Vol 32 ◽  
pp. 447-453
Author(s):  
Qi Kong ◽  
Ligong Wang
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Strongly Regular Graph ◽  
Upper Bounds ◽  
Free Graph ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Strongly Regular

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. The results are the following: (1) Let $1< k\leq l< \Delta < n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3\Delta+k-2l+1+\sqrt{(3\Delta+k-2l+1)^{2}+16l(\Delta+n-1)}$$ with equality if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$

The distance Laplacian and distance signless Laplacian spectrum of the subdivision-vertex join and subdivision-edge join of two regular graphs

2019 ◽  
Vol 11 (05) ◽  
pp. 1950053
Author(s):  
Deena C. Scaria ◽  
G. Indulal
Keyword(s):  
Regular Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Vertex Disjoint ◽  
Jahangir Graph

Let [Formula: see text] be a connected graph with a distance matrix [Formula: see text]. Let [Formula: see text] and [Formula: see text] be, respectively, the distance Laplacian matrix and the distance signless Laplacian matrix of graph [Formula: see text], where [Formula: see text] denotes the diagonal matrix of the vertex transmissions in [Formula: see text]. The eigenvalues of [Formula: see text] and [Formula: see text] constitute the distance Laplacian spectrum and distance signless Laplacian spectrum, respectively. The subdivision graph [Formula: see text] of a graph [Formula: see text] is obtained by inserting a new vertex into every edge of [Formula: see text]. We denote the set of such new vertices by [Formula: see text]. The subdivision-vertex join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. The subdivision-edge join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. In this paper, we determine the distance Laplacian and distance signless Laplacian spectra of subdivision-vertex join and subdivision-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. As an application we exhibit some infinite families of cospectral graphs and find the respective spectra of the Jahangir graph [Formula: see text].

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.

scholarly journals Construction for the Sequences of Q-Borderenergetic Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Bo Deng ◽  
Caibing Chang ◽  
Haixing Zhao ◽  
Kinkar Chandra Das
Keyword(s):  
Regular Graph ◽  
Laplacian Spectrum ◽  
New Class ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Arbitrary Graphs

This research intends to construct a signless Laplacian spectrum of the complement of any k-regular graph G with order n. Through application of the join of two arbitrary graphs, a new class of Q-borderenergetic graphs is determined with proof. As indicated in the research, with a regular Q-borderenergetic graph, sequences of regular Q-borderenergetic graphs can be constructed. The procedures for such a construction are determined and demonstrated. Significantly, all the possible regular Q-borderenergetic graphs of order 7<n≤10 are determined.

scholarly journals Critical group structure from the parameters of a strongly regular graph

2021 ◽  
Vol 180 ◽  
pp. 105424
Author(s):  
Joshua E. Ducey ◽  
David L. Duncan ◽  
Wesley J. Engelbrecht ◽  
Jawahar V. Madan ◽  
Eric Piato ◽  
...  
Keyword(s):  
Regular Graph ◽  
Group Structure ◽  
Strongly Regular Graph ◽  
Critical Group ◽  
Strongly Regular

scholarly journals Properties of Commuting Graphs over Semidihedral Groups

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 103
Author(s):  
Tao Cheng ◽  
Matthias Dehmer ◽  
Frank Emmert-Streib ◽  
Yongtao Li ◽  
Weijun Liu
Keyword(s):  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Vertex Connectivity ◽  
Laplacian Energy ◽  
Number Of Spanning Trees

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.

Sign in / Sign up

Export Citation Format

Share Document