ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P3.03
https://doi.org/10.26493/1855-3974.2882.c5e
(Also available at http://amc-journal.eu)
Comparing Wiener, Szeged and revised Szeged
index on cactus graphs
Stefan Hammer *
Graz University of Technology, Rechbauerstraße 12, Graz, Austria
Received 9 May 2022, accepted 1 November 2022, published online 11 January 2023
Abstract
We show that on cactus graphs the Szeged index is bounded above by twice the Wiener
index. For the revised Szeged index the situation is reversed if the graph class is further
restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index
is bounded below by twice its Wiener index. Additionally, we show that these bounds
are sharp and examine the cases of equality. Along the way, we provide a formulation of
the revised Szeged index as a sum over vertices, which proves very helpful, and may be
interesting in other contexts.
Keywords: Wiener index, (Revised) Szeged index, cactus graphs.
Math. Subj. Class. (2020): 05C09
1 Introduction
Presumably the first topological graph index, the Wiener index, was invented in 1947 by the
chemist Wiener [25], and is used to correlate physicochemical properties to the structure
of chemical compounds [3, 11]. Since then it was and still is thoroughly studied, see e.g.
[1, 4, 5, 6, 7, 17] for only some of the latest results. Over time many more topological graph
indices were devised and investigated. One such topological graph index is the Szeged
index that came up as an extension of a formula for the Wiener index of trees. It was first
introduced in [10] without proper name. By its construction it has meaningful connections
to the Wiener index. However, Randić found that the Szeged index is lacking something
for chemical applications in comparison to the Wiener index, and thus introduced in [21]
a slightly adapted variant of the Szeged index, the later so-called revised Szeged index.
It produces better correlations in chemistry than the normal Szeged index [21] and both
Szeged indices combined can be used to provide a measure of bipartivity of graphs [20].
*Stefan Hammer acknowledges the support of the Austrian Science Fund (FWF): W1230.
E-mail address: stefan.hammer@tugraz.at (Stefan Hammer)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 23 (2023) #P3.03
It is rather easy to see that the Wiener index and the (revised) Szeged index coincide
on trees. Furthermore, in 1994 some conjectures about the relation of the Wiener and the
Szeged index on connected graphs were made by Dobrynin and Gutman [8, 10]. A year
later already Dobrynin and Gutman proved that the Wiener index and the Szeged index are
equal if and only if every block of the graph is complete [9]. Another year later Klavžar
et al. showed that the Szeged index is at least as big as the Wiener index [16]. Since
then many more authors investigated the relation of the Wiener and the Szeged index, see
[2, 13, 14, 19] and references therein. This research has been extended to the revised Szeged
index [15, 29, 30] and to certain graph classes [12, 18, 24], with the most recent work on
cactus graphs dating from this year. For further current research in the context of comparing
graph indices with the Wiener index, we refer the interested reader to [26, 27, 28].
In this paper, we want to show new relations between Wiener, Szeged and revised
Szeged index for the special case of cactus graphs. Namely, we prove that the Szeged
index is bounded above by twice the Wiener index. In case of the revised Szeged index
the situation is more complex. For bipartite cacti the revised Szeged is equal to the Szeged
index, but if we limit the class of cactus graphs to those that have only cycles as blocks,
we can reverse the above statement. That is, the revised Szeged index is bounded below by
twice the Wiener index. Additionally, we show that these bounds are sharp and examine
the cases of equality. Along the way, we provide a formulation of the revised Szeged index
as a sum over vertices, which proves very helpful, and may be interesting in other contexts.
The paper is organized as follows. In Section 2, we first introduce the main definitions
and directly afterwards show how the revised Szeged index can be written as a sum over
vertices (Theorem 2.1). Then we introduce some auxiliary results needed in the following
sections. The relation of the Szeged index and the Wiener index on cactus graphs is the
main topic of Section 3. We show that the Szeged index is bounded above by twice the
Wiener index (Theorem 3.1), and also look at equality cases. Section 4 starts with an
example showing that arbitrary cactus graphs can have a revised Szeged index equal to
twice its Wiener index. As a consequence, we look at a subclass of the cactus graphs to
prove a reverse relation for the revised Szeged and the Wiener index (Theorem 4.2).
2 Preliminaries and the revised Szeged index as vertex sum
If not otherwise mentioned, we are working with a finite, simple and connected graph G,
that has vertex set V (G) and edge set E(G). Let u, v be vertices of G. Then we denote
with dG(u, v) the distance of u and v in G, that is, the length of the shortest path connecting
u and v in G. For a path P , we use |P | for its length. Furthermore, we write nG(u, v) for
the number of vertices closer to u than to v, and oG(u, v) = oG(v, u) for the number of
vertices with equal distance to u and v. With this, the Wiener index, the Szeged index, and
the revised Szeged index are defined respectively by
W (G) =
∑
{u,v}⊆V (G)
dG(u, v) =
1
2
∑
u,v∈V (G)
dG(u, v),
Sz(G) =
∑
{s,t}∈E(G)
nG(s, t)nG(t, s),
Sz∗(G) =
∑
{s,t}∈E(G)
(
nG(s, t) +
1
2
oG(s, t)
)(
nG(t, s) +
1
2
oG(t, s)
)
.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 3
Note, that the Wiener index is a sum over all unordered pairs of vertices, whereas the
(revised) Szeged index is a sum over all edges.
In [22], Simić et al. introduced for vertices u, v and an edge {s, t} the function
µu,v({s, t}) =

1 if

dG(u, s) < dG(u, t) and dG(v, s) > dG(v, t),
or
dG(u, s) > dG(u, t) and dG(v, s) < dG(v, t),
0 otherwise.
This can be considered an indicator function that is 1 if and only if the vertices u and v
contribute to nG(s, t)nG(t, s). Bonamy et al. [2] used µu,v to rewrite the Szeged index in
the following way:
Sz(G) =
∑
{u,v}⊆V (G)
∑
{s,t}∈E(G)
µu,v({s, t}).
With this reformulation, the Szeged index is also a sum over all unordered pairs of vertices.
Additionally, Bonamy et al. called all edges e satisfying µu,v(e) = 1, ‘good’ for {u, v}, and
referenced this again to Simić et al. [22]. However, Simić et al. used the term ‘good edge’
for a completely different concept. Because of this, and the fact that the term ‘good’ is not
descriptive, we decided to use a different notation. We call edges e satisfying µu,v(e) =
1, (u, v)-distance-disparate, and denote with disG(u, v) the number of (u, v)-distance-
disparate edges in G. Hence, we can write for the Szeged index,
Sz(G) =
∑
{u,v}⊆V (G)
disG(u, v) =
1
2
∑
u,v∈V (G)
disG(u, v).
Since the revised Szeged index may not even be an integer, there cannot be a single
indicator function as there is for the Szeged index. So it seems difficult to formulate the
revised Szeged index as sum over vertices. Still a rather similar approach works. The first
step is to consider an equivalent of µu,v for single vertices and edges having end points
with the same distance to the vertex. Namely, we define for a vertex v and an edge {s, t},
νv({s, t}) =
{
1 if dG(v, s) = dG(v, t),
0 otherwise,
an indicator function that is 1 if and only if the end points of the edge have the same distance
to v. Now, similar to before, we call edges e satisfying νu(e) = 1 and νv(e) = 1 for vertices
u and v, (u, v)-distance-equal, and denote with deqG(u, v) the number of (u, v)-distance-
equal edges in G. These are the ingredients necessary to write the revised Szeged index as
sum over vertices.
Theorem 2.1. The revised Szeged index of a graph G can be written as sum over vertices
in the following form:
Sz∗(G) =
1
2
∑
u,v∈V (G)
(
disG(u, v) + deqG(u, u)−
1
2
deqG(u, v)
)
.
4 Ars Math. Contemp. 23 (2023) #P3.03
Proof. Let n be the number of vertices in G. Use that n = nG(s, t) + nG(t, s) + oG(s, t)
for all edges {s, t} to rewrite the revised Szeged index:
Sz∗(G) =
∑
{s,t}∈E(G)
(
nG(s, t) +
1
2
oG(s, t)
)(
nG(t, s) +
1
2
oG(t, s)
)
=
∑
{s,t}∈E(G)
(
nG(s, t)nG(t, s) +
1
2
oG(s, t)(n− oG(s, t)) +
1
4
oG(s, t)
2
)
= Sz(G) +
1
2
n
∑
{s,t}∈E(G)
oG(s, t)−
1
4
∑
{s,t}∈E(G)
oG(s, t)
2.
(2.1)
Since a vertex v is counted in oG(s, t) if and only if dG(v, s) = dG(v, t), we can rewrite
the second sum to∑
{s,t}∈E(G)
oG(s, t) =
∑
u∈V (G)
∑
e∈E(G)
νu(e) =
∑
u∈V (G)
deqG(u, u).
For the third term notice that vertices u and v are involved in oG(s, t) · oG(s, t) if and only
if dG(u, s) = dG(u, t) and dG(v, s) = dG(v, t), that is {s, t} is counted in deqG(u, v).
Thus, we can reformulate this sum as well:∑
{s,t}∈E(G)
oG(s, t)
2 =
∑
u,v∈V (G)
deqG(u, v).
Insert the reformulations and the Szeged index written as vertex sum in Equation (2.1) and
write for n the sum over all vertices to get the desired result:
Sz∗(G) =
1
2
∑
u,v∈V (G)
disG(u, v) +
1
2
n
∑
u∈V (G)
deqG(u, u)−
1
4
∑
u,v∈V (G)
deqG(u, v)
=
1
2
∑
u,v∈V (G)
(
disG(u, v) + deqG(u, u)−
1
2
deqG(u, v)
)
.
A noteworthy consequence of the above result is that the difference between the Szeged
and the revised Szeged index can be nicely described.
Corollary 2.2. The difference between the Szeged and the revised Szeged index of a graph
G on n vertices satisfies
Sz∗(G)− Sz(G) = 1
2
∑
{s,t}∈E(G)
(
n · oG(s, t)−
1
2
oG(s, t)
2
)
=
1
2
n
∑
u∈V (G)
deqG(u, u)−
1
4
∑
u,v∈V (G)
deqG(u, v).
Before we come to the comparison of the Wiener index and the (revised) Szeged in-
dex on cactus graphs, we need some general results about graphs. The first is about the
connection of disG and dG on cycles.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 5
Lemma 2.3. Let u and v be two distinct vertices of a cycle C of length n. Then
disC(u, v) =
{
2 dC(u, v) if n is even,
2 dC(u, v)− 1 if n is odd.
Proof. To make things easier, we think of a suitable embedding of C in the plane and say
right for counterclockwise, and left for clockwise. For some vertex w in C, let Pr(w) be
the path starting at w and going ⌊n/2⌋ edges to the right, and Pl(w) the path starting at w,
going ⌊n/2⌋ edges to the left. We denote the terminal vertices of Pr(w) and Pl(w) with wr
and wl, respectively.
Let e be an edge in C. It is clear that if e is in Pr(u), then the left vertex of e is closer
to u, and vice versa, if e is in Pl(u), then the right vertex of e is closer to u. For v the
situation is the same. Thus, e is (u, v)-distance-disparate if and only if it is contained in the
path Pr(u) ∩ Pl(v), or in the path Pl(u) ∩ Pr(v).
Without lost of generality, we can assume v is in Pr(u), see Figure 1 for an exemplary
illustration of the situation. In this case, Pr(u) ∩ Pl(v) is a shortest path from u to v, and
Pl(u) ∩ Pr(v) is a shortest path from ul to vr. So we have
disC(u, v) = dC(u, v) + dC(ul, vr). (2.2)
By inclusion–exclusion principle, the distance from ul to vr can be determined by
dC(ul, vr) = |Pl(u) ∩ Pr(v)|
= |Pr(u) ∩ Pl(v)|+ |Pl(u)|+ |Pr(v)| − |E(C)|
= dC(u, v) + 2 ⌊n/2⌋ − n.
Now considering even and odd n respectively, and inserting dC(ul, vr) in (2.2) completes
the proof.
u
ur = ul
v
vr = vl
Pr(u)Pl(u)
Pr(v)
Pl(v) u
urul
v
vr
vl
Pr(u)Pl(u)
Pr(v)
Pl(v)
Figure 1: Cycle C of even length left and of odd length right, with vertices u, v, and the
paths going right and left including their terminal vertices.
The next result is about splitting distances in a block-cut-vertex decomposition of the
given graph. Recall, a block is a maximal 2-connected subgraph, and in a block-cut-vertex
decomposition blocks only overlap at cut vertices. More information on blocks and the
block-cut-vertex decomposition can be found in [23].
6 Ars Math. Contemp. 23 (2023) #P3.03
Proposition 2.4. Let u and v be vertices of a graph G with set of blocks B obtained by
the block-cut-vertex decomposition for G. For a block B in B, denote by uB and vB the
vertices in B closest to u and v, respectively. Then
dG(u, v) =
∑
B∈B
dG(uB , vB). (2.3)
Proof. Let P be a shortest path from u to v and B′ ⊆ B the set of blocks visited by
P . Every block B in B′ is entered by uB and left by vB , so P can be decomposed into
subpaths PB , where for a block B the subpath PB starts at uB and ends at vB . Since every
subpath of a shortest path is a shortest path itself, it follows that
dG(u, v) = |P | =
∑
B∈B′
|PB | =
∑
B∈B′
dG(uB , vB). (2.4)
Now consider a block B not visited by P . Since we have a block-cut-vertex decompo-
sition, there is a unique vertex w in B minimizing the distance from the block B to the path
P . This vertex is also a cut vertex and thus it minimises the distance from B to any vertex
of the path P . Hence, it follows that uB = vB = w, and dG(uB , vB) = 0. This finishes
the proof.
Note, in the proof of Proposition 2.4 we do not use that blocks are two-connected. That
means, instead of blocks, we could split the graph into arbitrary subgraphs that only overlap
at cut vertices.
In the remaining two sections, we apply the above tools to the so called cactus graphs.
These are connected graphs where every two distinct cycles have at most one common
vertex. Alternatively, the graph consists of a single vertex, or every block is either an edge
or a cycle.
3 Comparing Wiener and Szeged index on cactus graphs
Since every edge on a shortest path from u to v is clearly (u, v)-distance-disparate, formu-
lating the Szeged index as sum over vertices gives the first part of the following inequality:
W (G) ≤ Sz(G) ≤ Sz∗(G).
Already Simić et al. used the indicator function µu,v to show additionally that equality
holds in the first part if and only if every block of G is complete, see [22, Theorem 2.1].
The inequality of the second part is clear by definition, whereas equality holds if and only
if G is bipartite. This was shown by Pisanski and Randić, see [20, Theorem 1]. Besides, it
follows from Corollary 2.2.
Here, we want to show a different inequality, true for the special class of cactus graphs.
Theorem 3.1. Let G be a cactus graph, then
Sz(G) ≤ 2W (G),
with equality if and only if every block of G is a cycle of even length.
A special case of this result was already given in [18]. There, Li and Zhang showed
that Theorem 3.1 holds for unicyclic graphs.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 7
Proof. Let u, v be vertices in G and e be an edge in a block B. With uB as in Propo-
sition 2.4 every shortest path from e to u uses uB . The same is true for v and vB as
in Proposition 2.4, respectively. Thus e is (u, v)-distance-disparate if and only if it is
(uB , vB)-distance-disparate. Hence, with B as set of blocks, we can write
disG(u, v) =
∑
B∈B
disG(uB , vB). (3.1)
Suppose that every block is a cycle of even length. Then by Lemma 2.3 and Proposi-
tion 2.4,
Sz(G) =
1
2
∑
u,v∈V (G)
disG(u, v) =
1
2
∑
u,v∈V (G)
∑
B∈B
disG(uB , vB)
=
1
2
∑
u,v∈V (G)
∑
B∈B
2dG(uB , vB) =
∑
u,v∈V (G)
dG(u, v)
= 2W (G).
(3.2)
Now if there is at least one odd cycle C, then again by Lemma 2.3, there is a strict
inequality instead of the third equality in the above formula. Finally, if there is a block
consisting of only a single edge {s, t}, then disG(s, t) = 1 = dG(s, t), and thus also
Sz(G) < 2W (G).
Note, blocks consisting of two vertices connected with two edges considered as cycles
of length two can be allowed in Theorem 3.1. Clearly, this is not a characterisation of
graphs G satisfying Sz(G) ≤ 2W (G), since every complete graph Kn satisfies Sz(Kn) =
W (Kn). Unfortunately, it is also not a characterisation of graphs satisfying Sz(G) =
2W (G). Below, we give an example of a graph satisfying the equation that is not a cactus
graph.
Example 3.2. Let G consist of three paths of length two joined at their end points. Attach
on one side of the end points of the paths two edges by their end points and on the other
side three edges. See Figure 2 for an exemplary drawing. It can be checked via a computer,
or even easily by hand that
Sz(G) = 192 = 2 · 96 = 2W (G).
Figure 2: A bipartite non-cactus graph G satisfying Sz(G) = 2W (G).
8 Ars Math. Contemp. 23 (2023) #P3.03
By generalizing the graph in Example 3.2 to have k paths instead of only 3, more
example graphs satisfying the equality can be found. Not for every k a suitable number of
edges can be attached, but it seems there is no cap for k. The biggest example graph G
we found has 783 paths of length 2, 28 edges on one and 656 009 edges on the other side
attached. It satisfies
Sz(G) = 862 902 435 600 = 2 · 431 451 217 800 = 2W (G).
This suggests that also if the cyclomatic number, which is just |E(G)| − |V (G)| + 1 for
connected graphs, is large, Sz(G) = 2W (G) can still hold for non-cactus graphs.
4 Comparing Wiener and revised Szeged index on cactus graphs
From the last section, we can conclude that Sz∗(G) ≤ 2W (G) holds for bipartite cactus
graphs G. But in case of non-bipartite cactus graphs the situation becomes more compli-
cated. There are even cactus graphs G satisfying Sz∗(G) = 2W (G), where not every
block is a cycle of even length as the following example shows.
Example 4.1. Take a cycle of length 13, a cycle of length 11, six edges and join them at a
single vertex to obtain a cactus graph G, as depicted in Figure 3. It can be checked that
Sz∗(G) = 3636 = 2 · 1818 = 2W (G).
Figure 3: A cactus graph G satisfying Sz∗(G) = 2W (G).
With this in mind, it seems difficult to make any concrete statements about the con-
nection of the revised Szeged and the Wiener index in the case of cactus graphs. Hence,
we focused on a subclass of cactus graphs and found the following relation, which is in
contrast to Theorem 3.1.
S. Hammer: Comparing Wiener, Szeged and revised Szeged index on cactus graphs 9
Theorem 4.2. Suppose every block of a graph G is a cycle. Then
2W (G) ≤ Sz∗(G),
with equality if and only if every cycle in G has even length.
Note, clearly a graph where every block is a cycle is a cactus graph.
Proof. Let u, v be vertices in G and e be an edge in a block B. Again, we use the notation
of Proposition 2.4 with uB and vB for the vertices in B closest to u and v, respectively.
Since uB is on every shortest path from u to e, and the same is true for vB and v, it is
evident that deqB(u, v) = deqB(uB , vB). Furthermore, the set of blocks B of G induces a
partition of the edge set. Hence,
deqG(u, v) =
∑
B∈B
deqB(u, v) =
∑
B∈B
deqB(uB , vB).
Thus, with Theorem 2.1 we can formulate the revised Szeged index of G as
Sz∗(G) =
1
2
∑
u,v∈V (G)
(
disG(u, v) + deqG(u, u)−
1
2
deqG(u, v)
)
=
1
2
∑
u,v∈V (G)
∑
B∈B
(
disG(uB , vB) + deqB(uB , uB)−
1
2
deqB(uB , vB)
)
.
(4.1)
Next we distinguish two cases, whereby the second case has two sub-cases, to show
that for any vertices uB and vB in a block B,
2 dG(uB , vB) ≤ disG(uB , vB) + deqB(uB , uB)−
1
2
deqB(uB , vB). (4.2)
Case 1: Suppose that B is a cycle of even length. Then,
deqB(uB , uB) = 0 = deqB(uB , vB),
and by Lemma 2.3,
disG(uB , vB) = 2 dG(uB , vB).
Case 2: Suppose that B is a cycle of odd length.
Case 2.1: If uB ̸= vB , then
deqB(uB , uB) = 1, deqB(uB , vB) = 0,
and again by Lemma 2.3
disG(uB , vB) = 2 dG(uB , vB)− 1.
Case 2.2: If uB = vB , then
deqB(uB , uB) = 1 = deqB(uB , vB),
disG(uB , vB) = 0 = 2 dG(uB , vB).
10 Ars Math. Contemp. 23 (2023) #P3.03
So in Case 1 and Case 2.1, we have equality in (4.2), and in Case 2.2, (4.2) is fulfilled
with a strict inequality. Therefore by Proposition 2.4 and (4.1),
2W (G) =
1
2
∑
u,v∈V (G)
∑
B∈B
2 dG(uB , vB) ≤ Sz
∗(G),
with equality if and only if every cycle in G has even length.
ORCID iDs
Stefan Hammer https://orcid.org/0000-0002-8064-8733
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