For a simple graph G of order n, size m and with signless Laplacian eigenvalues q1,q2,…,qn, the signless Laplacian energy QE(G) is defined as QE(G)=∑i=1n|qi−d‾|, where d‾=2mn is the average vertex ...degree of G. We obtain the lower bounds for QE(G), in terms of first Zagreb index M1(G), maximum degree d1, second maximum degree d2, minimum degree dn and second minimum degree dn−1. As a consequence of these bounds, we obtain several bounds for the energy E(L(G)) of the line graph L(G) of graph G in terms of various graph parameters like M1(G), ω (the clique number), n, m, etc., which improve some recently known bounds.
Let
D
be a simple digraph with
n
-vertices,
m
arcs having skew Laplacian eigenvalues
ν
1
,
ν
2
,
⋯
,
ν
n
-
1
,
ν
n
=
0
. The skew Laplacian energy
S
L
E
(
D
)
of a digraph
D
is defined as
S
L
E
(
D
)
...=
∑
i
=
1
n
|
ν
i
|
. In this paper, we obtain the characteristic polynomial of skew Laplacian matrix of the digraph
D
1
→
D
2
and also obtain the
S
L
E
(
D
1
→
D
2
)
in terms of
S
L
E
(
D
1
)
and
S
L
E
(
D
2
)
and show the existence of some families of skew Laplacian equienergetic digraphs.
Let G be a simple connected graph with n vertices and m edges. Let W(G)=(G,w) be the weighted graph corresponding to G. Let λ1,λ2,…,λn be the eigenvalues of the adjacency matrix A(W(G)) of the ...weighted graph W(G). The energy E(W(G)) of a weighted graph W(G) is defined as the sum of absolute value of the eigenvalues of W(G). In this paper, we obtain upper bounds for the energy E(W(G)), in terms of the sum of the squares of weights of the edges, the maximum weight, the maximum degree d1, the second maximum degree d2 and the vertex covering number τ of the underlying graph G. As applications to these upper bounds we obtain some upper bounds for the energy (adjacency energy), the extended graph energy, the Randić energy and the signed energy of the connected graph G. We also obtain some new families of weighted graphs where the energy increases with increase in weights of the edges.
Let
D
be a simple connected digraph with
n
vertices and
m
arcs and let
W
(
D
)
=
(
D
,
ω
)
be the weighted digraph corresponding to
D
, where the weights are taken from the set of non-zero real ...numbers. In this paper, we define the skew Laplacian matrix
S
L
(
W
(
D
)
)
and skew Laplacian energy
S
L
E
(
W
(
D
)
)
of a weighted digraph
W
(
D
)
, which is defined as the sum of the absolute values of the skew Laplacian eigenvalues, that is,
S
L
E
(
W
(
D
)
)
=
∑
i
=
1
n
|
ρ
i
|
, where
ρ
1
,
ρ
2
,
…
,
ρ
n
are the skew Laplacian eigenvalues of
W
(
D
)
. We show the existence of the real skew Laplacian eigenvalues of a weighted digraph when the weighted digraph has an independent set such that all the vertices in the independent set have the same out-neighbors and in-neighbors. We obtain a Koolen type upper bound for
S
L
E
(
W
(
D
)
)
. Further, for a connected weighted digraph
W
(
D
)
, we obtain bounds for
S
L
E
(
W
(
D
)
)
, in terms of different digraph parameters associated with the digraph structure
D
. We characterize the extremal weighted digraphs attaining these bounds.
Skew Laplacian energy of digraphs Ganie, Hilal A.; Chat, Bilal A.
Afrika mathematica,
06/2018, Volume:
29, Issue:
3-4
Journal Article
Peer reviewed
In this paper, we consider the Laplacian energy of digraphs. Various approaches for the Laplacian energy of a digraph have been put forward by different authors. We consider the skew Laplacian energy ...of a digraph as given in Cai et al. (Trans Combin 2:27–37,
2013
). We obtain some upper and lower bounds for the skew Laplacian energy which are better than some previous known bounds. We also show every even positive integer is the skew Laplacian energy of some digraphs.
The set of all non-increasing non-negative integer sequences $pi=(d_1, d_2,ldots,d_n)$ is denoted by $NS_n$. A sequence $piin NS_{n}$ is said to be graphic if it is the degree sequence of a ...simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $pi$. The set of all graphic sequences in $NS_{n}$ is denoted by $GS_{n}$. The complete product split graph on $L + M$ vertices is denoted by $overline{S}_{L, M}=K_{L} vee overline{K}_{M}$, where $K_{L}$ and $K_{M}$ are complete graphs respectively on $L = sumlimits_{i = 1}^{p}r_{i}$ and $M = sumlimits_{i = 1}^{p}s_{i}$ vertices with $r_{i}$ and $s_{i}$ being integers. Another split graph is denoted by $S_{L, M} = overline{S}_{r_{1}, s_{1}} veeoverline{S}_{r_{2}, s_{2}} vee cdots vee overline{S}_{r_{p}, s_{p}}= (K_{r_{1}} vee overline{K}_{s_{1}})vee (K_{r_{2}} vee overline{K}_{s_{2}})vee cdots vee (K_{r_{p}} vee overline{K}_{s_{p}})$. A sequence $pi=(d_{1}, d_{2},ldots,d_{n})$ is said to be potentially $S_{L, M}$-graphic (respectively $overline{S}_{L, M}$)-graphic if there is a realization $G$ of $pi$ containing $S_{L, M}$ (respectively $overline{S}_{L, M}$) as a subgraph. If $pi$ has a realization $G$ containing $S_{L, M}$ on those vertices having degrees $d_{1}, d_{2},ldots,d_{L+M}$, then $pi$ is potentially $A_{L, M}$-graphic. A non-increasing sequence of non-negative integers $pi = (d_{1}, d_{2},ldots,d_{n})$ is potentially $A_{L, M}$-graphic if and only if it is potentially $S_{L, M}$-graphic. In this paper, we obtain the sufficient condition for a graphic sequence to be potentially $A_{L, M}$-graphic and this result is a generalization of that given by J. H. Yin on split graphs.
Recognition of split-graphic sequences Chat, Bilal A.; Pirzada, Shariefudddin; Iványi, Antal
Acta Universitatis Sapientiae. Informatica,
12/2014, Volume:
6, Issue:
2
Journal Article
Peer reviewed
Open access
Using different definitions of split graphs we propose quick algorithms for the recognition and extremal reconstruction of split sequences among integer, regular, and graphic sequences.
We consider the skew Laplacian matrix of a digraph
G
→
obtained by giving an arbitrary direction to the edges of a graph G having n vertices and m edges. We obtain an upper bound for the skew ...Laplacian spectral radius in terms of the adjacency and the signless Laplacian spectral radius of the underlying graph G. We also obtain upper bounds for the skew Laplacian spectral radius and skew spectral radius, in terms of various parameters associated with the structure of the digraph
G
→
and characterize the extremal graphs.
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