The weighted version of the handshaking lemma with an application
Journal of Inequalities and Applications volume 2014, Article number: 351 (2014)
The purpose of the present note is to establish the weighted version of the handshaking lemma with an application to chemical graph theory.
The graphs under consideration are finite, but may have loops and parallel edges. Let be a graph. For a vertex , denotes the degree of the vertex v, that is, the number of edges incident with v, where a loop incident with v is counted twice. We say that and are the order and size of G, respectively.
Around the middle of the 20th century theoretical chemists recognized that useful information on the dependence of various properties of organic substances on molecular structure can be obtained by examining pertinently constructed invariants of underlying molecular graphs. Eventually, graph invariants that are useful for chemical purpose, were named ‘topological indices’ or ‘molecular structure-descriptors’. A large number of various ‘topological indices’ was proposed and studied in chemical literature .
In 1975, Randić  introduced the so-called ‘branching index’. The branching index of a graph G is now widely known as the Randić index of G, defined as
We refer to the monograph  and the survey article  for the various results on the Randić index, and to [5–11] for some recent results concerning Randić index of graphs. Bollobás and Erdős  showed that for a graph of order n without isolated vertices, with equality if and only if G is the star . The sharp upper bound for the Randić index of graphs of order n is due to Fajtlowicz .
Theorem 1.1 (Fajtlowicz )
For a graph G of order n,
with equality if and only if every component of G is regular and G has no isolated vertices.
The proof of Fajtlowicz is based on Cauchy’s inequality. Caporossi et al.  gave an alternative proof of Theorem 1.1 by using an equivalent formulation for the Randić index of a graph:
where is the number of isolated vertices in G. Using linear programming, Pavlović and Gutman  gave another proof of Theorem 1.1. In the present note, we give a short proof of Theorem 1.1, based on the weighted version of the handshaking lemma, which reads as follows.
Theorem 1.2 (The weighted version of the handshaking lemma)
Let f be any complex valued function defined on the vertex set of a graph G. Then
By letting for each vertex in the above lemma, one can deduce the handshaking lemma.
Corollary 1.3 (The handshaking lemma)
For any graph G of size m,
By taking the function f in Theorem 1.2 as
Corollary 1.4 For any graph G of order n,
where is the number of isolated vertices in G.
Let denote the set of isolated vertices in a graph G. Došlic et al. established an identity similar to Theorem 1.2, which reads as follows.
Lemma 1.5 (Došlic et al. )
holds for any graph G and any function f.
Note that Theorem 1.2 is an extension of Lemma 1.5. Although both Corollary 1.4 and the following identity for the first Zagreb index (see [17, 18] for some recent results on this parameter) of a graph G can also be deduced from Lemma 1.5:
there are some which cannot be deduced from Lemma 1.5 but can be deduced from Theorem 1.2. For instance,
where is the eccentricity of u in G, i.e., the largest distance between u and any other vertex v of G. We remark that
is defined by Ghorbani and Hosseinzadeh in , while the eccentric connectivity index
is defined by Sharma et al. .
Another application of Theorem 1.2 is a new expression of the connective eccentricity index of a connected graph G, defined by Yu and Feng  as
By Theorem 1.2,
2 The proofs
Proof of Theorem 1.2 Every term occurs times in the left hand side summation, as it occurs exactly the same times in the right hand side of the identity. □
Proof of Theorem 1.1 It is well known that for any two positive real numbers a and b, their geometric mean is greater than or equal to their harmonic mean, that is,
with equality if and only if . Therefore, together with Corollary 1.4,
where is the number of isolated vertices in G. If , we conclude from the above inequalities that and for any two adjacent vertices u and v in G. It follows that G has no isolated vertices and every component of G is regular.
Next assume that G has no isolated vertices, be all components of G, and be a -regular graph of order for any . Then
by the handshaking lemma (Corollary 1.3), whence
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The author is grateful to the referees for their helpful comments and to Professor Fajtlowicz for kindly sending a copy of ‘Written on the Wall, a list of conjectures of Graffiti’. This work was supported by NSFC (No. 11161046).
The author declares to have no competing interests.