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 Some Graphs Determined by their Signless Laplacian (Distance) Spectra

2020 ◽  
Vol 36 (36) ◽  
pp. 461-472
Author(s):  
Chandrashekar Adiga ◽  
Kinkar Das ◽  
B. R. Rakshith
Keyword(s):  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Discrete Math ◽  
Complete Graphs ◽  
Spectral Determination ◽  
Regular Graphs ◽  
Bipartite Subgraph ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

In literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtained from the complete graph $K_{n}$ by removing all the edges of a complete bipartite subgraph $K_{l,m}$ is studied. It is shown that the graph $K_{n}\backslash K_{1,m}$ with $m\ge4$ is determined by its signless Laplacian spectrum, and it is proved that the graph $K_{n}\backslash K_{l,m}$ is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph $K_{n-2\alpha}\vee \alpha K_{2}$ was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em Linear Algebra Appl.}, 436:3634--3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for $n-2\alpha=1 ~\text{or}~ 2$. Here, this problem is completely solved for all positive integer $n-2\alpha$. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.

scholarly journals Spectral determination of some chemical graphs

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1123-1131 ◽  
Author(s):  
Changjiang Bu ◽  
Jiang Zhou ◽  
Hongbo Li
Keyword(s):  
The Other ◽  
Spectral Determination ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Chemical Graphs ◽  
Signless Laplacian

Let Tk denote the caterpillar obtained by attaching k pendant edges at two pendant vertices of the path Pn and two pendant edges at the other vertices of Pn. It is proved that Tk is determined by its signless Laplacian spectrum when k = 2 or 3, while T2 by its Laplacian spectrum.

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.

Spectrum of graphs obtained by operations

2018 ◽  
Vol 13 (02) ◽  
pp. 2050045
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Somnath Paul
Keyword(s):  
Distance Matrix ◽  
Laplacian Matrix ◽  
Regular Graphs ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Simple Connected Graph ◽  
Equienergetic Graphs

The distance signless Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text]. In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, determined by the graph [Formula: see text]. As an application, we show that new pairs of auxiliary equienergetic graphs can be constructed by joined union of regular graphs.

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 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 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.

The spectra of a new join of graphs

2018 ◽  
Vol 10 (06) ◽  
pp. 1850074 ◽  
Author(s):  
Somnath Paul
Keyword(s):  
Laplacian Spectrum ◽  
Cospectral Graphs ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Join Of Graphs ◽  
Disjoint Sets

Let [Formula: see text] and [Formula: see text] be three graphs on disjoint sets of vertices and [Formula: see text] has [Formula: see text] edges. Let [Formula: see text] be the graph obtained from [Formula: see text] and [Formula: see text] in the following way: (1) Delete all the edges of [Formula: see text] and consider [Formula: see text] disjoint copies of [Formula: see text]. (2) Join each vertex of the [Formula: see text]th copy of [Formula: see text] to the end vertices of the [Formula: see text]th edge of [Formula: see text]. Let [Formula: see text] be the graph obtained from [Formula: see text] by joining each vertex of [Formula: see text] with each vertex of [Formula: see text] In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of [Formula: see text] in terms of those of [Formula: see text] and [Formula: see text] As an application, we construct infinite pairs of cospectral graphs.

scholarly journals On Q-integral (3,s)-semiregular bipartite graphs

2010 ◽  
Vol 4 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Slobodan Simic ◽  
Zoran Stanic
Keyword(s):  
Bipartite Graphs ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. We establish some general results regarding signless Laplacians of semiregular bipartite graphs. Especially, we consider those semiregular bipartite graphs with integral signless Laplacian spectrum. In some particular cases we determine the possible Q-spectra and consider the corresponding graphs.

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