Applicable Analysis and Discrete Mathematics
available online at http://p efmath.etf.rs
Appl. Anal. Discrete Math. 4 (2010), 156–166. doi:10.2298/AADM1000001C
TOWARDS A SPECTRAL THEORY OF GRAPHS
BASED ON THE SIGNLESS LAPLACIAN, III
Dragoˇs Cvetkovi´c
1
, Slobodan K. Simi´c
1
This part of our work further extends our project of building a new spectral
theory of graphs (based on the signless Laplacian) by some results on graph
angles, by several comments and by a short survey of recent results.
1. INTRODUCTION
This is the third part of our work with a common title. The first [11] and the
second part [12] will be also referred in the sequel as Part I and Part II, respectively.
This third part was not planned at the beginning, but a lot of recently pub-
lished papers on the signless Laplacian eigenvalues of graphs and some observations
of ours justify its preparation.
By a spectral graph theory we understand, in an informal sense, a theory
in which graphs are studied by means of eigenvalues of a matrix M which is in
a prescribed way defined for any graph. This theory is called M–theory. Hence,
there are several spectral graph theories (for example, those based on the adjacency
matrix, the Laplacian, etc.). In that sense, the title “Towards a spectral theory
of graphs based on the signless Laplacian” indicates the intention to build such a
spectral graph theory (the one which uses the signless Laplacian without explicit
involvement of other graph matrices).
Recall that, given a graph, the matrix Q = D + A is called the signless
Laplacian, where A is the adjacency matrix and D is the diagonal matrix of vertex
degrees.
In fact, we outlined in [11], [12] a new spectral theory of graphs (based on
the signless Laplacian Q). We shall call this theory the Q–theory.
2000 Mathematics Subject Classification. 05C50.
Keywords and Phrases. Graph theory, graph spectra, adjacency matrix, Laplacian, signless Lapla-
cian.
1
Supported by the Serbian Ministry of Science and Technological Development, grant 144015G
156
Towards a spectral theory of graphs based on the signless Laplacian, III 157
We have also compared the Q-theory with other spectral theories, in particu-
lar to those based on the adjacency matrix A and the Laplacian L. As demonstrated
in the first part, the Q–theory can be constructed in part using various connections
to other theories: equivalence with A–theory and L–theory for regular graphs, com-
mon features with L–theory for bipartite graphs, general analogies with A–theory
and analogies with A–theory via line graphs and subdivision graphs. In Part I we
also presented results on graph operations, inequalities for eigenvalues and recon-
struction problems. In Part II we introduced notions of enriched and restricted
spectral theories and presented results on integral graphs, enumeration of span-
ning trees, characterizations by eigenvalues, cospectral graphs and graph angles.
This part further extends our project by some results on graph angles, by several
comments and by a short survey of recent results.
We use here the terminology and notation from Parts I and II although a
part of that we repeat here.
Only recently has the signless Laplacian attracted the attention of researchers.
As our bibliography shows, several papers on the signless Laplacian spectrum have
been published since 2005 and we are now in a position to summarize the develop-
ments. In the first part of this paper we have mentioned 15 papers (in particular,
[4], [6], [10], [15], [20], [22], [23], [35], [39], [41], [43], [44], [46], [47], [55], where
the signless Laplacian is used explicitly) in addition to our previous basic papers
[5], [9]. In Part II we have added the following 11 references: [1], [3], [24], [25],
[26], [27], [34], [36], [48], [52], [54]. In the meantime the following 16 papers [2],
[13], [18], [19], [21], [28], [29], [30], [31], [32], [38], [42], [45], [49], [51], [53] have
been published or are in the process of publication at this moment. Together with
[11], [12] and this paper, there are in this moment about 50 papers on the signless
Laplacian spectrum published since 2005. Several others are forthcoming.
The rest of the paper is organized as follows. Section 2 surveys the progress
in resolving some conjectures on the signless Laplacian eigenvalues which are gen-
erated by computer. Recent results on spectral characterizations are presented in
Section 3. The largest eigenvalue is the subject of Section 4. We present in Section
5 new results related to graph angles. Other results are noted in Section 6. Section
7 contains some concluding remarks.
2. RESOLVING CONJECTURES
Paper [10] is devoted to inequalities involving Q–eigenvalues. It presents
30 computer generated conjectures in the form of inequalities for Q–eigenvalues.
Conjectures that are confirmed by simple results already recorded in the literature,
explicitly or implicitly, are identified. Some of the remaining conjectures have
been resolved by elementary observations; for others quite a lot of work had to be
invested. The conjectures left unresolved appear to include some difficult research
problems. One difficult conjecture (Conjecture 24) has been confirmed in [4] by a
long sequence of lemmas.
Conjecture 25 appears also to be a difficult one. It remains unsolved but
158 Dragoˇs Cvetkovi´c, Slobodan K. Simi´c
some related work is described in Section 6.
Conjectures 6, 7 and 10 from [10] have been proved in [26].
Crucial to the resolution of these conjectures was the following result related
to largest Q-eigenvalue q
1
(G) of a graph G.
Theorem 1. Let G be a connected graph with n vertices and m edges. Then
q
1
(G) ≤
2m
n − 1
+ n − 2,
with equality if and only if G is K
1,n−1
or K
n
.
The inequality of Theorem 1 is better than our bound in Theorem 3.4 of Part
I. The two bounds are equal only for complete graphs. The best upper bound for
q
1
in terms of n and m is implicitly given by Theorems 3.2 and 3.3 of Part I.
In order to prove Theorem 1, the authors of [26] derive first the bound
(1) q
1
(G) ≤ max{d
v
+ m
v
|v ∈ V (G)},
where d
v
is the degree of the vertex v and m
v
the average degree of neighbors of
v. As noted in Part I, paper [35] checks whether known upper bounds on largest
Laplacian eigenvalue µ
1
hold also for q
1
and establishes that many of them do
hold, in particular inequality (1). However, the authors of [35] claim that (1) was
implicitly proved in [16].
To complete the proof of Theorem 1 the authors of [26] use another inequality
by K. Ch. Das [17]:
max{d
v
+ m
v
|v ∈ V (G)} ≤
2m
n − 1
+ n − 2.
Some results related to Conjecture 7 can be found in [1].
Theorems 3.5 and 3.6 of Part I confirm Conjecture 19 and 20 of [10], respec-
tively.
The question of equality in Theorem 3.6. (a = q
2
) remained unresolved in
Part I. Graphs for which equality holds are among the graphs with λ
3
= 0. To this
group belong the graphs mentioned in Conjecture 20 of [10] (stars, cocktail–party
graphs, complete bipartite graphs with equal parts). We can add here regular com-
plete multipartite graphs in general (cocktail–party graphs and complete bipartite
graphs with equal parts are special cases).
Paper [18] settled completely the question of equality in Theorem 3.6 of Part
I (Conjecture 20).
The same paper confirmed the lower bound of Conjecture 14 together with
Conjectures 15, 22 and 23. This was achieved using a lower bound for the second
largest Q-eigenvalue and an upper bound for the least Q-eigenvalue in terms of
vertex degrees.
At the moment the following conjectures of [10] remain unconfirmed: those
parts related to upper bounds in Conjectures 8, 9, 11, 14 together with Conjectures
16, 17, 18, 21, 25, 26.
Towards a spectral theory of graphs based on the signless Laplacian, III 159
The paper [2] discusses the same set of conjectures and presents some new
ones.
A new set of conjectures involving the largest Q–eigenvalue appears in [31].
The Q-index is considered in connection with various structural invariants, such
as diameter, radius, girth, independence and chromatic number, etc. Out of 152
conjectures, generated by computer (i.e. by the system AGX), many of them
are simple or proved in [31], so that only 18 remained unsolved. An additional
conjecture of this type has been resolved in [32]; it is proved that q
1
(G) ≤ 2n(1 −
1/k), where k is the chromatic number, thus improving an analogous inequality for
the A-index (cf. [7], p. 92).
3. SPECTRAL CHARACTERIZATIONS
In Part II we had the following paragraph.
Starlike trees are DS in the L–theory [37], while this is not proved for the A–theory [50].
Concerning the Q–theory, a private communication of Omidi is cited in [14] by which T –shape
trees (starlike trees with maximal degree equal to 3) are DS except for K
1,3
. We can verify this
assertion by reducing the problem via subdivision graphs to A–theory and then using results of
[50]. Indeed, the subdivision graph of a T –shape tree is again a T –shape tree and an A–cospectral
mate, described in [50], is not a subdivision graph except for K
1,3
.
Recently the paper [38] has appeared. Contrary to his previous private com-
munication, mentioned above, G. R. Omidi proves now that not only K
1,3
but
an infinite series of T -shape trees which are not DS does exist. When confirming
the original private communication we made a mistake. The mistake was that the
A–cospectral mate, mentioned above, is still a subdivision graph yielding the Q-
cospectral mate found in [38]. Hence, our method of using subdivision graphs and
results from [50] do confirm the results of [38]. In fact our method proves these
results in a much simpler way.
The paper [38] provides an infinite series of pairs of Q-cospectral graphs, one
graph in each pair being bipartite and the other non-bipartite. The only such pair
of Q-cospectral graphs previously noted in the literature consists of the graphs K
1,3
and C
3
∪ K
1
.
Assume that G is not DS. We shall say that G is minimal graph which is not
determined by its spectrum if removing of any subset of its components implies
that the remaining graph is DS. In what follows, only the minimal graphs which
are not DS will be considered, since any other such graph can be easily recognized
by the presence of minimal graphs.
We consider the class of graphs whose each component is either a path or a
cycle. We shall classify the graphs from the considered class into those which are
determined, or not determined, by their spectrum.
For signless Laplacian spectra the problem is implicitly solved in [12] (see
Subsection 3.3, Theorem 2.9 and the example after it) and explicitly in [13]. It
follows that C
2k
∪2P
`
and C
3
∪K
1
are minimal non DS graphs. Using subdivisions
160 Dragoˇs Cvetkovi´c, Slobodan K. Simi´c
of graphs (which reduces the problem to usual spectrum), and having in mind
relations between the spectra, one can see that no other minimal non–DS graphs
exist. Moreover, these considerations solve also the problem for the set of graphs
whose largest signless Laplacian eigenvalue does not exceed 4. The only additional
non-DS graph is K
1,3
which is cospectral to C
3
∪ K
1
.
As shown in [13], where A-, L- and Q-eigenvalues are considered, in the
class of graphs whose each component is a path or a cycle, the cospectrality as a
phenomenon the most rarely appears in the case of signless Laplacian spectrum.
Graphs consisting of two cycles with just a vertex in common are called ∞-
graphs in [49]. It is proved that ∞-graphs without triangles are characterized by
their Laplacian spectra and that all ∞-graphs, with one exception, are characterized
by their signless Laplacian spectra.
4. THE LARGEST EIGENVALUE
The study of the largest Q-eigenvalue remains an attractive topic for re-
searchers. In particular, the extremal values of the Q-index in various classes of
graphs, and corresponding extremal graphs, have been investigated.
In [24] the class of unicyclic graphs with a given number of pendant vertices
or given independence number was considered. Graphs with maximal Q-index and
corresponding extremal graphs are determined.
Independently, the same results have been obtained in [53], in a more general
setting. Graphs with maximal Q-index in the class of graphs with given vertex
degrees are determined and these results are applied to unicyclic graphs.
In [21] the class of bicyclic graphs with a given number of pendant vertices
was considered. Graphs with maximal Q-index and corresponding extremal graphs
are determined.
A graph G is a quasi-k-cyclic graph if it contains a vertex (say r, the root
of G) such that G − r is a k-cyclic graph, i.e. a connected graph with cyclomatic
number k (= m − n + 1, where n is the number of vertices and m is the number
of edges). For example, if k = 0, the corresponding graph is a quasi-tree. In [28]
quasi-k-cyclic graphs having the largest Q-index are identified for k ≤ 2.
Explicit expression for the characteristic polynomial of the signless Laplacian
of a nested split graph (or threshold graphs) in terms of vertex degrees is derived
in [51].
Recall that the total graph of G, denoted by T (G), is the graph with vertex
set corresponding to union of vertex and edge sets of G, with two vertices of T (G)
adjacent if the corresponding elements in G are adjacent or incident. It is also
well known that T (G) = S(G)
2
(see [33]), where S(G) is a subdivision of G, while
square stands for the 2-power graph (so H
2
has the same vertex set as H, with two
vertices being adjacent if their distance in H is ≤ 2). The above relation implies
that
Q(T (G)) = A
2
(S(G)) + Q(S(G)),