For a connected graph , the edge Mostar index is defined as , where and are respectively, the number of edges of lying closer to vertex than to vertex and the number of edges of lying closer to vertex than to vertex . We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that achieve the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95–106].
Let be a graph with vertex set
and edge set . The order and size of are the cardinality of and , respectively. The distance between
and in , denoted by , is the shortest path connecting
and in . For a vertex and edge of a graph , the
distance between and , denoted by , is defined as .
A molecular graph is a simple graph such that its vertices correspond
to the atoms and the edges to the bonds of a molecule. A topological
graph index, also called a molecular descriptor, is a mathematical
formula that can be applied to any graph which models some molecular
structure. From this index, it is possible to analyse mathematical
values and further investigate some physicochemical properties of a
molecule. Therefore, it is an efficient method in avoiding expensive and
time-consuming laboratory experiments.
Došlić et al. [7]
introduced the Mostar index, which is a measure of peripherality in
graphs. It can also be considered as a bond-additive structural
invariant as a quantitative refinement of the distance non-balancedness.
For a graph , the Mostar index is
defined as where is the number of vertices of
closer to than to and is the number of vertices closer
to than to .
Since its introduction in 2018, the Mostar index has already incited
a lot of research, mostly concerning trees [1,6,5,4,8,12,14], unicyclic graphs [18], bicyclic graphs [20], tricyclic graphs [11], cacti [13] and chemical graphs [3,16,21,22].
Arockiaraj et al. [2],
introduced the edge Mostar index as a quantitative refinement of the
distance non-balancedness, also measure the peripherality of every edge
and consider the contributions of all edges into a global measure of
peripherality for a given chemical graph. The edge Mostar index of is defined as
where , and
are respectively, the
number of edges of lying closer
to vertex than to vertex and the number of edges of lying closer to vertex than to vertex . We use for short, if
there is no ambiguity.
The problem of determining which graphs uniquely maximize (resp.
minimize) the edge Mostar index in various classes of graphs has
received much attention. For example, Imran et al. [17] studied the edge Mostar index
of chemical structures and nanostructures using graph operations. Havare
[10] computed the edge Mostar
index for several classes of cycle-containing graphs. Hayat et al. [15] obtained a sharp upper bound
for the edge Mostar index on tricyclic graphs and identified the graphs
that attain the bound. Liu et al. [19] determined the extremal values of the edge
Mostar index among trees, unicyclic graphs, cacti and posed two
conjectures for the extremal edge Mostar index among bicyclic graphs.
Ghalavand et al. [9] gave
proof to a conjecture in [19], and obtained the graph that minimizes the edge
Mostar index among bicyclic graphs.
Motivated directly by [19]
and [9], we determine the
unique graph that maximizes the edge Mostar index over all bicyclic
graphs with fixed size, which disproves the following conjecture.
Conjecture 1.1.
[19] If the size
of bicyclic graphs is large enough, then the bicyclic graphs (see Figure 1) and (see Figure 2) have the maximum
edge Mostar index.
We disprove Conjecture 1.1, by presenting the following result.
Theorem 1.2.
Let be a bicyclic graph of size . Then (Where and are depicted in Figure 1 and Figure 2,
respectively).
Based on the above Theorem, if the size of bicyclic graphs is large, then is the unique graph that maximizes
the edge Mostar index. Therefore, Conjecture 1.1 is
disproved.
In Section 2, we provide some definitions and preliminary
results. The proof of Theorem 1.2 is presented in Section 3.
2. Preliminaries
In this section, we present some basic notations and elementary
results, which will be useful in the proof of our main result.
Figure 1 The graphs , and , each of size , have , , and pendent edges, respectively
Figure 2 The graphs , and of size , have , , , and pendent edges,
respectively
For , let be the set of vertices that are
adjacent to in . The degree of , denoted by , is the cardinality of . A vertex with degree one is
called a pendent vertex and an edge incident to a pendent vertex is
called a pendent edge. A graph
with vertices is a bicyclic graph
if . As usual by , , and we denote the star, path, and cycle
graph on vertices,
respectively.
The join of two graphs and
is denoted by , is obtained by identifying
one vertex from and . Let be the identified vertex in both and . If contains a cycle and belongs to some cycle, while is a tree, then we call a pendent tree in associated with .
For each , every
path from to some edges in pass through . Therefore, the contribution of to totally depends
on the size of . In other words,
changing the structure of does
not affect the value of .
Lemma 2.1.
[19] Let be a graph
of size with a cut edge . Then with equality if and only
if is a pendent edge.
By Lemma 2.1, if
is a pendent edge, then is
maximum. Therefore, it is easy to verify the following result.
Lemma 2.2.
Let be a graph of size . Then where the common vertex of and is the center of , while the common vertex of and can be chosen arbitrarily.
Let ,
where the common vertex of
and is the center of .
Let be a connected graph of size and be a unicyclic graph of size . Then where the common vertex of and (resp. and ) is the center of (resp. ), while the common vertex of
and can be chosen arbitrarily.
Proof. If , then by Lemma 2.3, we have
Similarly, if ,
then ; if , then .
3. Proof of theorem 1.2
The -graph is a graph that
connects two fixed vertices with three internally disjoint paths. If the
lengths of these paths are , and where , the graph is
denoted as .
Let represent
the set of bicyclic graphs of size that contain exactly two edge-disjoint
cycles. Let denote
the set of bicyclic graphs of size that contain exactly three cycles.
Moreover, if ,
then contains a -graph as a subgraph, which is
called as the brace of .
For the proof of Theorem 1.2, we first develop several lemmas. In the
first step, we establish a sharp upper bound for on the set .
Lemma 3.1.
Let . Then
Proof. Let . Then there are two unicyclic graphs and with size and , respectively such that . Then, in view of Lemma
2.4. If
, we get if ,
then
If , we have
By simple calculation, we get , , and . If , then
. This completes the proof.
In the following, we determine a sharp upper bound for on the set .
Lemma 3.2.
Let with brace . Then
Proof. Let with brace . Suppose that be the vertices in of with , and . Let be the number of pendent edges of
(). Suppose that and . Let be the graph obtained from by shifting pendent edges from to . We have
Hence,
and equality holds if and only if , i.e., .
Let be the graph obtained
from by shifting pendent edges from to . We have
Hence, and equality holds if and only if , i.e., .
Let be the graph obtained
from by shifting pendent edges from to . We have
Hence, and equality holds if and only if , i.e., . Clearly, with , and . Also, by simple
calculation, we have , .
Lemma 3.3.
Let with brace . If , then with equality if
and only if (see
Figure 2).
Proof. Let with brace . Let be vertices in of with and . Let be the number of pendent edges of
(). Suppose that and . Let be the graph obtained from by shifting pendent edges from to . We have
Hence,
and equality holds if and only if , i.e., .
Let be the graph obtained
from by shifting pendent edges from to . We get
Hence, and equality holds if and only if , i.e., .
Let be the graph obtained
from by shifting pendent edges from to . By symmetry, , and equality
holds if and only if .
Let be the graph obtained from
by shifting pendent edges from to . We get
Hence, and equality holds if and only if , i.e., . Note that . Clearly, .
Lemma 3.4.
Let with brace . If , then with equality if
and only if (see
Figure 2).
Proof. Let with brace . Let be vertices in of with and . Let be the number of pendent edges of
(). Suppose that . Let be the graph obtained from by shifting (resp. ) pendent edges from (resp. ) to (resp. ). We deduce that
Hence,
and equality holds if and only if , i.e., .
Let be the graph obtained
from by shifting pendent edges from to . We obtain
Hence, and equality holds if and only if , i.e., .
Let be the graph obtained
from by shifting pendent edges from to . We have
Hence, and equality holds if and only if , i.e., . Note that . Clearly, .
Lemma 3.5.
Let with brace . Then
Proof. Suppose that
and are the vertices with degree
3 in of . Let be the three
disjoint paths connecting and
. We now proceed by considering
the following three possible cases.
Case 1..
Subcase 1.1..
We choose six edges such that each one is incident to or in . Let be one of
these six edges. Then . This property holds for the remaining five edges as well.
Therefore,
Subcase 1.2..
Let be one of the four edges
incident to or in the paths and , and be one of the two edges incident to
or in the path . Then , and . Hence,
Case 2..
Subcase 2.1..
The Subcase follows from Lemma 3.3.
Subcase 2.2..
Let be one of the four edges
in the paths and , and be one of the two edges incident to
or in the path . Then , and . Hence,
Subcase 2.3..
Let be one of the four edges
incident to or in the paths and , and is an edge in the path . Then , and . Hence,
Case 3..
Subcase 3.1..
By Lemma 3.2 and simple calculation, we have if , then ; if , then ; if , then ; if , then
; if , ; if ,
then .
Subcase 3.2..
This Subcase follows from Lemma 3.4.
Subcase 3.3..
Let , be one of the four edges in the paths
and that are incident to or , and be one of the two edges in the middle
of the path . Then , , and . Thus,
Subcase 3.4..
Let , be one of the two edges in the path
that are incident to or , be one of the two edges in the path
that are incident to or , and be one of the two edges in the middle
of the path . Then , , , and . Hence,
The proof of Theorem 1.2 directly follows from Lemmas 3.1 and 3.5.
Acknowledgements
The authors express their gratitude to the anonymous referee for
their insightful comments and helpful suggestions. This work was
supported by the National Natural Science Foundation of China (Grant
Nos. 12071194, 11571155 and 12071158).
Declarations
The authors declare no conflict of interest.
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