In this article, we study the resolvent energy and the pseudospectrum energy of a sequence
of fullerenes with exactly 12n carbon atoms. In particular, the resolvent energy of these fullerenes ...together with their lower bounds are obtained.
Laplacian energy of a graph Gutman, Ivan; Zhou, Bo
Linear algebra and its applications,
04/2006, Volume:
414, Issue:
1
Journal Article
Peer reviewed
Open access
Let
G be a graph with
n vertices and
m edges. Let
λ
1,
λ
2,
…
,
λ
n
be the eigenvalues of the adjacency matrix of
G, and let
μ
1,
μ
2,
…
,
μ
n
be the eigenvalues of the Laplacian matrix of
G. An ...earlier much studied quantity
E
(
G
)
=
∑
i
=
1
n
|
λ
i
|
is the
energy of the graph
G. We now define and investigate the
Laplacian energy as
LE
(
G
)
=
∑
i
=
1
n
|
μ
i
-
2
m
/
n
|
. There is a great deal of analogy between the properties of
E(
G) and
LE(
G), but also some significant differences.
Degree-based energies of graphs Das, Kinkar Ch; Gutman, Ivan; Milovanović, Igor ...
Linear algebra and its applications,
10/2018, Volume:
554
Journal Article
Peer reviewed
Let G=(V,E) be a simple graph of order n and size m, with vertex set V(G)={v1,v2,…,vn}, without isolated vertices and sequence of vertex degrees Δ=d1≥d2≥⋯≥dn=δ>0, di=dG(vi). If the vertices vi and vj ...are adjacent, we denote it as vivj∈E(G) or i∼j. With TI we denote a topological index that can be represented as TI=TI(G)=∑i∼jF(di,dj), where F is an appropriately chosen function with the property F(x,y)=F(y,x). A general extended adjacency matrix A=(aij) of G is defined as aij=F(di,dj) if the vertices vi and vj are adjacent, and aij=0 otherwise. Denote by fi, i=1,2,…,n the eigenvalues of A. The “energy” of the general extended adjacency matrix is defined as ETI=ETI(G)=∑i=1n|fi|. Lower and upper bounds on ETI are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases.
On the energy of line graphs Akbari, Saieed; Alazemi, Abdullah; Anđelić, Milica ...
Linear algebra and its applications,
03/2022, Volume:
636
Journal Article
Peer reviewed
The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) 3 it was conjectured that for every graph G with ...maximum degree Δ(G) and minimum degree δ(G) whose adjacency matrix is non-singular, E(G)≥Δ(G)+δ(G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E(G). In this paper, first we show that E(L(G))≥|E(G)|+Δ(G)−5, where L(G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7.
Let G be a graph. The energy of G,
, is defined as the sum of absolute values of its eigenvalues. Here, it is shown that if G is a graph and
is an edge partition of G, such that
are spanning; then
if ...and only if
, for every
and
, where
is the adjacency matrix of
. It was proved that if G is a graph and
are subgraphs of G which partition edges of G, then
. In this paper we show that if G is connected, then the equality is strict, that is
.
More on borderenergetic graphs Deng, Bo; Li, Xueliang; Gutman, Ivan
Linear algebra and its applications,
05/2016, Volume:
497
Journal Article
Peer reviewed
Open access
The energy E(G) of a graph G is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. If a graph G of order n has the same energy as the complete graph Kn, i.e., if ...E(G)=2(n−1), then G is said to be borderenergetic. We obtain three asymptotically tight bounds on the edge number of borderenergetic graphs. Then, by using disconnected regular graphs we construct connected non-complete borderenergetic graphs.
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