On the bounds for signless Laplacian energy of a graph

Ganie, Hilal A.; Pirzada, S.

doi:10.1016/j.dam.2016.09.030

Cited by 19 publications

(12 citation statements)

References 22 publications

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“…(1) does not hold. This can be seen from the proof of Lemma 5 (Theorem 3.3 in [22]). In order to make the equality in Lemma 5 hold, on the one hand, graph G must have only one signless Laplacian eigenvalue greater than or equal to average degree 2m n , that is, there should hold q 2 < 2m n for G. On the other hand, for the line graph ℓ(G) of G, its largest adjacency eigenvalue should satisfy µ 1 (ℓ(G)) = 2m(ℓ(G)) n(ℓ(G)) , that is, the line graph ℓ(G) of G is regular, (see page 55, theorem 3.2.1 in [23]).…”

Section: Lemma 5 ([22]mentioning

confidence: 82%

-borderenergetic graphs

Tao

Hou

2020

AKCE International Journal of Graphs and Combinatorics

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A graph G is said to be borderenergetic (L-borderenergetic, respectively) if its energy (Laplacian energy, respectively) equals the energy (Laplacian energy, respectively) of the complete graph K n. We extend this concept to signless Laplacian energy of a graph. A graph G is called Q-borderenergetic if its signless Laplacian energy is same as that of the complete graph K n , i.e., Q E(G) = Q E(K n) = 2n − 2. In this paper, we construct some infinite family of graphs satisfying Q E(G) = L E(G) = 2n − 2, this happens to give a positive answer to the open problem mentioned by Nair Abreu et al. in Nair Abreu et al. (2011), that is whether there are connected non-bipartite, non-regular graphs satisfying Q E(G) = L E(G). We also obtain some bounds on the order and size of Q-borderenergetic graphs. Finally, we use a computer search to find out all Q-borderenergetic graphs on no more than 10 vertices, the number of such graphs is 39.

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Section: Lemma 5 ([22]mentioning

confidence: 82%

-borderenergetic graphs

Tao

Hou

2020

AKCE International Journal of Graphs and Combinatorics

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“…is the first Zagreb index proposed by Gutman and Trinajstić [7]. The other two lower bounds for QE(G) in Corollary 3.2 and Theorem 3.1 in [6] are…”

Section: Introductionmentioning

confidence: 93%

“…This equation is an extension of the concept of graph energy. Similar to the Laplacian energy, the signless Laplacian energy of a graph G as put forward by Ganie, Hilal and Pirzada [6] is defined as QE(G) = n i=1 |q i − 2m n |. Particularly, if G is a regular graph, then…”

Section: Introductionmentioning

confidence: 99%

Some new bounds for the signless Laplacian energy of a graph

Wang,

Huang

2020

Preprint

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For a simple graph G with n vertices, m edges and signless Laplacian eigen-n is the average vertex degree of G. In this paper, we obtain two lower bounds ( see Theorem 3.1 and Theorem 3.2 ) and one upper bound for QE(G) ( see Theorem 3.3 ), which improve some known bounds of QE(G), and moreover, we determine the corresponding extremal graphs that achieve our bounds. By subproduct, we also get some bounds for QE(G) of regular graph G.

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“…Among the pioneering results of the theory of graph energy are the lower and upper bounds for energy, see [2,5,15,16,18,19,22,26] and the references therein. For more information about energy of graph see [1,9,10,11,12,14,23] and related results see [1,24,25]. A subset S of the vertex set V (G) is said to be a covering set of G if every edge of G is incident to at least one vertex in S. A covering set with minimum cardinality among all covering sets is called the minimum covering set of G and its cardinality, which is denoted by τ = τ (G) is called the vertex covering number of the graph G. If H is a subgraph of the graph G, we denote the graph obtained by removing the edges in H from G by G \ H (that is, only the edges of H are removed from G).…”

Section: Introductionmentioning

confidence: 99%

Bounds for graph energy in terms of vertex covering and clique numbers

Samee¹,

Pirzada

Ganie

2019

ejgta

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Let G be a simple graph with n vertices, m edges and having adjacency eigenvalues λ 1 , λ 2 ,. .. , λ n. The energy E(G) of the graph G is defined as E(G) = n i=1 |λ i |. In this paper, we obtain the upper bounds for the energy E(G) in terms of the vertex covering number τ , the clique number ω, the number of edges m, maximum vertex degree d 1 and second maximum vertex degree d 2 of the connected graph G. These upper bounds improve some of the recently known upper bounds.

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On the bounds for signless Laplacian energy of a graph

Cited by 19 publications

References 22 publications

-borderenergetic graphs

-borderenergetic graphs

Some new bounds for the signless Laplacian energy of a graph

Bounds for graph energy in terms of vertex covering and clique numbers

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