1. Introduction
In what follows, a graph
will be a finite simple connected undirected graph with vertex set
, edge set E and degrees
. A graph is d-regular if all its vertices have degree d; a graph is
-regular if its vertices have degree either a or b. Sometimes we will call these latter graphs semiregular, when the values of a and b are not needed explicitly. As we emphasized in [1] , as opposed to some other articles on the indices under study, we do not require that the semiregular graphs be bipartite. For all graph theoretical terms the reader is referred to reference [2] .
In Mathematical Chemistry, molecules are modeled using these graphs, where the vertices are the atoms and the atomic bonds are represented by the edges. Many topological indices, or descriptors, i.e., real-valued functions on the domain of all graphs, have been defined with the purpose of capturing physico-chemical properties of the molecules and classifying them according to the values of their indices. One such index is the symmetric division deg index, defined by
(1)
for any graph G, and introduced by Vukičević and Gašperov in [3] as one of the 148 so-called Adriatic indices. This index, which has a good predictive power for the total surface area of polychlorobiphenyls, has been studied in a number of articles, of which we mention [1] [4] [5] and [6] where additional references can be found.
In [7] we worked with a resistive relative of the SDD index, the mixed degree-Kirchhoff index defined as
(2)
where
is the effective resistance between vertices i and j when the graph is thought of as an electrical network, where all the edges have unit resistance. In that article we found the relation
that was used extensively in [1] and rediscovered in [8] .
Ghorbani et al. introduced in [8] the inverse symmetric division deg index, defined as
that uses the inverses of the quotients used in (1), and proved, among other things, a lower bound for the product of the SDD and ISDD indices, as well as lower bounds for the difference of the indices. Some optimal results for the ISDD index are found in [9] for unicyclic graphs, and in [10] for trees and chain graphs.
In this article we want to continue working with the interplay between the two indices proving an upper bound for their product, and studying the large family of graphs for which the equalities are attained. Additionally, we give two new upper bounds for the ISDD index using known lower bounds for the SDD index found in the literature.
2. The Results
For a general descriptor of the form
where the cis are non-negative parameters of G, and for which
, we found in [11] the following bounds:
Lemma 1. For any descriptor
we have
(3)
Both equalities are attained in case
for all
. Also, the right equality is attained if n is even and the first
of the
s are equal to m and the other
of the
s are equal to M.
Also, if n is odd we have
(4)
Both equalities are attained in case
for all
. Also, the right equality is attained if the first
of the
s are equal to m, the last
of the
s are equal to M, and the middle
is either m or M.
The left inequalities are a consequence of the arithmetic-harmonic-mean inequality. The right inequalities are shown using Schweitzer inequality, originally found in [12] , and Lupaş inequality, a refinement of Schweitzer inequality
proved in [13] . Let us denote by
the quotients
, for
, and let
,
. Applying the lemma we have the following:
Theorem 1. For all G we have
(5)
where both equalities are attained if all quotients are equal, and the right equality is also attained attained if
is even, and
of the quotients are equal to m and the other
are equal to M.
Also, if
is odd, we have
(6)
where the equality is attained if all quotients
are equal, or if
of them are equal to m and the rest are equal to M, or if
of them are equal to m and the rest are equal to M.
We want to give sufficient conditions for the cases where the equalities in (5) and (6) are attained. As Ghorbani et al. noticed ( [8] , thm. 4), the quotients are all equal if the graph is regular or edge-transitive.
Regularity and edge-transitivity are sufficient but not necessary conditions for the right equalities of (5) and (6) to be attained: the natural condition here, other than regularity, is
-regularity. Obviously edge-transitivity guarantees
-regularity for some k and r, but the opposite is not necessarily true. Consider for example the graph built from two 4-cycles, or squares, S1 and S2, such that one vertex v1 of S1 is linked to another vertex v2 of S2 with a 2-path graph P2, and the diagonally opposite vertices of v1 and v2 are linked with another P2. Then this is a 10-vertex
-regular graph which is not edge-transitive, because not all edges are part of a square. Still, not even
-regularity is necessary. A more general condition would be that for all pairs of edges
and
, there is a positive integer q such that
and
. For example, the 9-vertex graph with incidence matrix
is neither edge-transitive nor
-regular. Two edges have degrees 1 and 2 and eight edges have degrees 2 and 4, so that all quotients satisfy
Now let us look at a family of graphs that have exactly two different quotients, and attain the equalities in (5) and (6). For
, consider the squares
,
, and link S1 with S2 with a P4 and for all other
, link
to
with a P8, using always diagonally opposite vertices for the linking. It is easy to see that the total number of
edges is equal to the total number of
edges, which is
. Replacing the P4 with a P3 produces a graph
where
of the edges have quotient m and the other
edges have quotient M; finally, replacing the P4 with a P5 produces a graph where
of the edges have quotient M and the other
edges have quotient m.
Thus, these are examples of graphs where the right equalities in (5) and (6) are attained, in all three possible cases, and all conditions of theorem 1 are nonempty. As a final application, we give a couple of upper bounds for the ISDD index using known lower bounds for the SDD index found in the literature.
Theorem 2. For any graph G with p pendent vertices and minimum non-pendent vertex degree
we have
(7)
where the equality holds for any regular graph and for the star graph.
Proof. Use theorem 3.3 in [6] and (5) in our Theorem 1.
In the case that G is regular, all quotients are equal to
, and thus
; on the other hand, on account of the facts that
and
, the right side also becomes
.
In the case that G is the star graph, all quotients are equal to
, and thus
which is equal to the value of the bound, on account of the facts that
and
The first and second Zagreb indices M1 and M2 were defined in [14] and [15] , respectively, as
and
With these indices we have the following:
Theorem 3. For any graph G we have
(8)
where the equality holds for any regular or semiregular graphs.
Proof. Use theorem 3.1 in [4] and (5) in our Theorem 1.
In the case that G is d-regular,
. It is easy to see that this is also the value of the bound on the right side, because
,
and
.
In the case that G is
-regular,
. Since
, the bound on the right becomes
(9)
but
and inserting this result into (9) gives us the same value for the upper bound obtained for