Bond Additive Modeling 5 . Mathematical Properties of the Variable Sum Exdeg Index Damir Vuki č evi ć

Recently, discrete and variable Adriatic indices have been introduced and it has been shown that the sum α -exdeg index is good predictor (when variable parameter is equal to 0.37 ) of the octanol-water partition coefficient for octane isomers. Here, we study mathematical properties of this descriptor. Namely, we analyze extremal graphs of this descriptor in the following classes: the class of all connected graphs, the class of all trees, the class of all unicyclic graphs, the class of all chemical graphs, the class of all chemical trees, the class of all chemical unicyclic graphs, the class of all graphs with given maximal degree, the class of all graphs with given minimal degree, the class of all trees with given number of pendant vertices, and the class of all connected graphs with given number of pendant vertices. Also, many open problems about variable Adriatic indices are proposed. (doi: 10.5562/cca1667)


INTRODUCTION
Recently, discrete and variable Adriatic indices have been introduced 1 and studied.Predictive and mathematical properties of discrete Adriatic indices have been analyzed in papers. 1,2Predictive properties of variable Adriatic indices have been studied in Ref. 3. It has been found that three variable Adriatic indices have especially good predictive properties, namely: 1) the inverse sum -1.95-deg index is well correlated with standard enthalpy of formation of octane isomers   2 0.75 R  2) the inverse sum 0.43-lodeg index is well correlated with total surface area of octane isomers   2 0.92 R  3) the sum 0.37-exdeg index is well correlated with the octanol-water partition coefficient   2 0.99 .R  In this paper, we restrict our attention to the variable sum exdeg index.The variable sum exdeg index is defined by: where   E G is the set of edges of .G Note that this index can be rewritten as: where   V G is the set of vertices of .G Hence, this index can be considered as a sum of vertex contributions such that the contribution of each vertex depends solely on its degree.One immediately see a parallel to the well known the first Zagreb index 4 defined by: The mathematical and predictive properties of Zagreb index have been extensively studied (see Refs. 4-6 and references within and, for recent mathematical studies, Refs.7-12).
In this paper, we analyze graphs with extremal values of the a SEI index in the following classes: the class of all connected graphs, the class of all trees, the class of all unicyclic graphs, the class of all chemical graphs, the class of all chemical trees, the class of all chemical unicyclic graphs, the class of all graphs with given maximal degree, the class of all graphs with given minimal degree, the class of all trees with given number of pendant vertices, and the class of all connected graphs with given number of pendant vertices.
These results can be used for the detection of chemical compounds that may have desirable properties.Namely, if one finds some property well-correlated with this descriptor for some value of , α then extremal graphs should correspond to molecules with minimal or maximal value of that property.Since one such property has already been found, 3 this may encourage the further study of this index.

a 
In this paper, we consider only simple connected graphs, so from now on by graph we imply simple connected graph.The number of vertices of G will be denoted by   n G and number of edges by  .
where 0. a  It can be easily seen that: From the Lemma 1, it directly follows that: for every graph G with n vertices and for each

 
for every graph G with n vertices and minimal degree ; δ and for each  x  and that 3, From here, it directly follows that: for each tree G with n vertices and for each 1. a  The equality for the lower bound holds if and only if G is a path.The equality holds for the upper bound holds if and only if G is a star .n S Proof: Note that 1 1 Let us recall that every graph G contains a spanning tree , T i.e. subgraph T which is a tree such that    .
V G V T  Let us prove: Let G be graph and T its spanning tree, then for each 1. a  Proof: Since the degree of every vertex in T is not larger then in , G the Lemma follows.■ From Proposition 8 and Lemma 9, it follows: for each graph G with n vertices and for each 1. a  The equality for the lower bound holds if and only if G is a path.Corollary 10.
for each chemical graph G with n vertices and for each 1. a  The equality for the lower bound holds if and only if G is a path.

Denote by
x are greater then 1.
Let us prove: The equality holds for the lower bound if and only if x x  and the equality for the upper bound .
n Without loss of generality, we may assume that 1 2 ... .
It can be easily seen that 1 1 x  and that 3, that it has at least 3 elements larger then 1, but this is not possible.Now, let us prove the upper bound.It can be easily seen that equality holds for i.e. that it has less then 3 elements larger then 1.This is possible only if 1 2 3 ...
S  be the graph obtained from the star n S by adding an edge connecting two leaves as presented in the Figure 1.From Lemma 11, it directly follows that:  S  by replacing some edges by paths.Similarly as above, it can be proved that: From this Lemma, it follows: for every tree T with n vertices and k pendant vertic- es, and for each Moreover, the equality in the lower bounds holds if and only if one of the following holds: and G is a path ; and G only has vertices of degrees 1 and 3; and G only has and G only has vertices of degrees and the inequality for the lower bound holds if and only if: Moreover, the equality for the upper bound holds if and only if conditions 1)-3) hold and the equality for the lower bound holds if and only if conditions I)-III) hold.
In order to prove the theorem, we need to find Now, we write the table of (approximate) values of the function at zero points and some arbitrary points (one smaller than all of them, one between each two successive zero points and one larger than all of them).(see Table 1) From this table ( Completely analogously, it can be proved that: and G be a unicyclic chemical graph with n vertices.It holds: Moreover, the equality in the lower bounds holds if and only if one of the following holds: and G is a cycle ; and G only has vertices of degrees 1 and 3; and G only has and G only has vertices of degrees This is a contradiction.■

CONCLUSIONS
In this paper we have analyzed extremal properties of the variable sum exdeg index  .
First, let us prove the lower bound.It can be easily seen that equality holds for  

S
be a class of graphs with n vertices obtained from 1 k

aSEI G
We have found the graphs with the extremal graphs in the following classes of graphs (with given number of vertices):1) class of all connected graphs 2) class of all trees 3) class of all unycyclic graphs 4) class of all chemical graphs 5) class of all chemical trees 6) class of all chemical unycyclic graphs 7) class of all graphs with given maximal degree 8) class of all graphs with given minimal degree 9) class of all trees with given number of pendant vertices 10) class of all connected graphs with given number of pendant our attention to chemical graphs, chemical trees and chemical unycyclic graphs.We leave the solution of the analogous problem in the remaining seven classes as an open problem.Further, we propose solving the analogous set of problems for the two descriptors that have shown good predicitve properties in paper Ref. 3. Namely, we propose the study of the following variable descriptors:  variable inverse sum deg index: propose the study of the generalizations of discrete Adriatic indices that have shown good predictive properties in paper Ref. 1.The problem regarding these generalizations are extensions of the open problems presented in paper Ref. 1. Namely, we propose the study of the following variable descriptors:  variable Randić type lodeg index: .Acta 84 (2011) 93. variable Randić type di index: 1. a  Equality holds if and only if G is a complete graph.Equality holds if and only if G is a  -regular graph.
every unicyclic graph G with n vertices and for each 1. a  E quality for the lower bound holds if and only if G is a cycle .
every graph G with n vertices and maximal degree , , .nkG S  ■Using similar techniques as above, it can be proved that:Proposition 15.It holds that for every graph G with n vertices and k pendant ver- tices, and for each 1. a  The equality for the lower bound holds if and only if G is a graph with k vertices The equality holds for the upper bound if and only if all pendant vertices are adjacent to the same vertex, and the subgraph obtained by elimination of pendant vertices is a complete graph □ The equality holds for the lower bound holds if and only if G is a path.The equality holds for the upper bound if and only if G has only vertices of degree 1 and 4. The equality holds for the lower bound holds if and only if G is a cycle.The equality holds for the upper bound if and only if G has only vertices of degree 1 and 4.

Table 1 .
Function values in significant points