The weighted version of the handshaking lemma with an application
© Wu; licensee Springer 2014
Received: 19 June 2014
Accepted: 3 September 2014
Published: 16 September 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 .
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 )
with equality if and only if every component of G is regular and G has no isolated vertices.
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)
By letting for each vertex in the above lemma, one can deduce the handshaking lemma.
Corollary 1.3 (The handshaking lemma)
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.
is defined by Sharma et al. .
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. □
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.
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).
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