Abstract
As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition, we establish upper and lower bounds for the Laplacian Estrada index of Г(G) based on the vertex degrees of G. These bounds are also connected with the number of spanning trees in Г(G).
1 Introduction
In this paper we are concerned with simple connected (molecular) graphs. Let G be such a graph with the vertex set V(G) = {υ1,…, υn} and the edge set E(G). The adjacency matrix of G is
The subdivided-line graph Г(G) of G is the line graph of the barycentric subdivision of G, namely, Г(G) = L(
The subdivided-line graph Г(G), combining both notions of line graph and barycentric subdivision, generalizes the class of Sierpiński-like graphs. Various structural properties, such as edge-disjoint Hamilton cycles, hamiltonian-connectivity, hub sets, connected dominating sets, independent spanning trees, and book-embeddings, have been systematically investigated in [5].
Among numerous graph-theoretic concepts, spanning trees have found a wide range of applications in mathematics, chemistry, physics and computer sciences. Denote by τ(G) the number of spanning trees in G. Enumeration of spanning trees in graphs with certain symmetry and fractals has been widely studied via ad hoc techniques capitalizing on the particular structures [6–11]. In general, we often have to resort to Kirchhoff’s celebrated matrix-tree theorem [12], which asserts that nτ(G) equals the product of all nonzero eigenvalues of Laplacian matrix of G, i.e.,
The Laplacian Estrada index of a (molecular) graph G with n vertices is defined as [15]
It is a close relative of the so-called Estrada index put forward by Estrada [16] in 2000, which has already found extensive applications in chemistry and physics. Many properties of LEE, including upper/lower bounds and extremal graphs, have been established (see e.g. [15, 17–20]). Here, to deal with LEE(Г(G)), we first derive bounds for the largest and second smallest eigenvalues λ1(Г(G)) and λ|Г(G))|-1(Г(G)). Based on these estimates and the obtained exact expression for τ(Г(G)), we manage to present upper and lower bounds for LEE(Г(G)) in terms of some basic graph parameters of G, including degrees and the number of edges.
2 Preliminaries
To begin with, we briefly review the electrically equivalent transformation technique introduced in [14].
An edge-weighted graph G (with the weight function
where
– Serial edges transformation: If two serial edges with conductances a and b are merged into a single edge with conductance
– Mesh-star transformation: If a complete graph Kt (t ≥ 2) with conductance a on all its edges is changed into a star K1, t with conductance ta on all its edges, we have
Fig. 1 shows an example of the above electrically equivalent transformations.
The following two lemmas on the Laplacian eigenvalues will be used in our proofs.
([21]) Let G be a simple graph. Then
If G is connected then the equality holds if and only if G is bipartite semiregular. Here, a semiregular graph G = (V, E) is a graph with bipartition (V1, V2) of V such that all vertices in Vi have the same degree ki for i = 1,2.
([22]) Let G be a simple connected graph with n ≥ 2 vertices and m edges. Then
with equality if and only if G is a complete graph.
To conclude this section, we present an inequality which will be instrumental in bounding LEE(Г(G)) later. It is also interesting in its own right.
Given an integer n ≥ 1 and a sequence a1 ≥ a2 ≥ … ≥ an ≥ 0, we have
The equality holds if a1 = … = an.
Notice that
3 Number of spanning trees related to degree sequence
The main result in this section is the following exact formula for the number of spanning trees in Г(G) in terms of the degree sequence of G.
Let G be a simple connected graph. Then
where
First, recall that a vertex of a graph is said to be pendant if its neighborhood contains exactly one vertex. The edge incident to a pendant vertex is called a pendant edge. Let
See Fig. 2 for an illustration (the purple node is inserted to subdivide the pendant edge).
To proceed, we will need the following edge-weighted graphs [23]. Define H′ as the edge-weighted version of
It is critical to observe that, as electrical networks, H″ can be obtained from H‴ by performing a series of mesh-star transformations taking each vertex v ∈ V(
where m := |E(G)| since each vertex in V(
Since H′ can be obtained from H″ by applying a series of serial edges transformations, we have
where the second equality holds since each edge in
again by noting that each edge in
Now, combining (5), (6), and (7) with (4), we have
In view of (2), we obtain
Hence, we readily obtain the expression (3) for τ(Г(G)) by plugging (9) into (8). The proof is complete.
The electric network technique for enumeration of spanning trees is particular useful when the graph in question has a high degree of symmetry; see e.g. [10] for an application on pseudofractal networks. It is worth noting that we do not assume any symmetry in G.
As a simple example, note that the graph G in Fig. 2 contains four spanning trees. Direct calculation using Theorem 3.1 yields τ(Г(G)) = 23. This is in line with the outcome from the matrix-tree theorem.
4 Bounds for Laplacian eigenvalues
We begin with the following upper bound for the largest Laplacian eigenvalue of a subdivided-line graph.
Let G be a simple connected graph. Then
where Δ(G) is the maximum degree of G. The equality holds if and only if G is a regular bipartite graph.
For each edge {u, v} ∈ E(Г(G)), the vertices u and v correspond to two incident edges, say, {u1, u2}, {u2, u3}, in
and
holds.
We consider two situations. (1) If
Thanks to Lemma 2.1, we obtain
with equality if and only if G is regular bipartite.
The first Zagreb index [24] of a graph G is defined as
Let G be a simple connected graph with | V(G)| ≥ 2. Then
where 2m = 2|E(G)| = | V(Г(G))|. The equality holds if and only if G is a single edge.
We know that |V(Г(G))| = |E(
Therefore, we readily arrive at (11) by employing Lemma 2.2.
We now discuss the sharpness of (11). If G is a single edge, then Г(G) = G. Lemma 2.2 implies that the equality holds in (11). Conversely, if the equality holds in (11), it follows from Lemma 2.2 that Г(G) must be a complete graph. But this is true only if G is a single edge. (Indeed, if G is not a single edge, G must contain a 2-path P2. Clearly, there are two vertices in Г(P2) that are not adjacent, and hence Г(G) cannot be complete.)
In [26], Mohar showed that
Hence,
5 Bounds for Laplacian Estrada index
In the light of the matrix-tree theorem which relates the Laplacian eigenvalues to the number of spanning trees, we in this section convert the above obtained results into bounds of the Laplacian Estrada index LEE(Г(G)).
Let G be a simple connected graph with |V(G)| > 2. Then
where 2m = 2|E(G)| = |V(Г(G))|. In the first inequality, equality holds if and only if G is a single edge, while the second equality holds if G is a single edge.
Furthermore,
with equality if G is a single edge.
By (1) and 2m = 2|E(G)| = |V(Г(G))|,
where we have used the fact that Г(G) is connected (and hence λ2m (Г(G)) = 0).
Recall that the matrix-tree theorem tells us that
where the equality holds if and only if Г (G) is a complete graph, which is again equivalent to the condition that G is a single edge (see the proof of Theorem 4.2).
For the second inequality in (13), we need to resort to Lemma 2.3. Similarly, we have
where the equality holds if G is a single edge.
The last statement concerning the inequality (14) follows by applying Theorem 4.1 and Theorem 4.2 to (13).
To show the availability of Theorem 5.1, we still use the graph G depicted in Fig. 2 as an example. Direct calculation shows
Acknowledgement
The author is grateful to the anonymous reviewers for their helpful comments and suggestions toward improving the original version of the paper. The author acknowledges support from the National Natural Science Foundation of China (11505127), the Shanghai Pujiang Program (15PJ1408300), and the Program for Young Excellent Talents in Tongji University (2014KJ036).
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