EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5943
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Proximity Prestige of a Vertex in Some Graph Families

Lysandra A. Toladro1,∗, Isagani S. Cabahug, Jr.2

1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University,
Musuan, Maramag, Bukidnon, 8710 Philippines

Abstract. Let G = (V,E) be an undirected graph where V,E are the set of vertices and edges
respectively. The proximity prestige (PP ) of a vertex vi is the sum of the shortest path distance
between vertex vi and vj all over the number of vertices in the graph. Proximity prestige (PP )
emphasizes the importance of both reachability and distance. Here, general properties of proximity
prestige in some classes of graph, including path, cycle, complete, friendship, complete bipartite,
star, fan and wheel were determined.

2020 Mathematics Subject Classifications: 05C12, 91D30

Key Words and Phrases: Proximity prestige, distance, graph families

1. Introduction

Graph theory has become an essential tool for analyzing complex systems and networks,
providing insights into the structure and dynamics of various real-world systems, from so-
cial networks to biological systems. One of the key aspects of graph analysis is centrality,
which aims to identify the most important or influential nodes in a network. Traditional
centrality measures, such as degree centrality, betweenness centrality, and closeness cen-
trality, have been widely used to capture different aspects of node importance based on
direct and indirect connections within a network. However, as networks grow increasingly
complex, these conventional measures may fail to fully capture the nuanced roles that
certain nodes play in facilitating information flow and influencing others.

A critical aspect of network analysis is centrality, which represents the importance of
a node by its position in the network. Using this type of information about social net-
work, Linton Freeman in 1976 [1] first proposed a measure of prestige called proximity
prestige. This measure, introduced by Freeman in 1970’s, considers not only the number
of connections a node has but also how accessible it is to others in the network. Proximity
prestige emphasizes strategically positioned nodes, by providing a perspective on influ-
ence and importance that goes beyond connectivity. This focuses on reach offers a more

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5943

Email addresses: lysatoladro@gmail.com (L. Toladro),
isaganicabahugjr@cmu.edu.ph (I. Cabahug, Jr.)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 2 of 13

complex conception of influence than is commonly used in disciplines like sociology and
organizational studies.

In 2015, these foundational studies by Freeman [1] and Zhao [2] motivated the current
investigation into proximity prestige. In this paper, the researcher’s employed the concept
of proximity prestige to study the prestige of vertices in nontrivial, connected, and undi-
rected graphs utilizing the results established by Zhao et al. [2]. By extending their work,
this study contributes to a deeper understanding of how proximity prestige can be used
to evaluate node importance in various types of networks, particularly in settings where
indirect influence is a critical factor.

This paper explores the formal definition of proximity prestige as it applies to vertices
in a graph, providing a mathematical framework for calculating node importance based
on their indirect connections. Other studies that deal with the concept of centralities
are located in [3] and [4]. For graph-theoretic terminologies not specifically defined nor
described in this study,please refer to either [5] or [6]. Therefore, all graphs considered in
this study are nontrivial, connected and undirected.

2. Terminology and Notation

2.1. Preliminary Concepts

A graph G is a finite nonempty set V of objects called vertices together with a
possibly empty set E of 2-element sets of V called edges. To indicate that a graph G has
vertex set V and edge set E, we write G = (V,E). To emphasize that V and E are the
vertex set and edge set of a graph G, we often write V as V (G) and E as E(G). Each
edge {u, v} of G is usually denoted by uv or vu. The number of vertices in a graph G is
the order of G and the number of edges is the sizeof G. The degree of a vertex v in
a graph G is the number of edges incident with v and is denoted by deg v or simply by
deg v. The degree of a vertex v is denoted by deg(v) and the minimum degree of G is
denoted by δ(G) and the maximum degree of G is denoted by ∆(G) [7].

If uv is an edge of G, then u and v are adjacent vertices. Two adjacent vertices
are referred to as neighbors of each other. The set of neighbors of a vertex v is called
the open neighborhood of v (or simply the neighborhood of v) and is denoted by NG(v)
or N(v) if the graph is understood. The set N [v] = N(v) ∪ {v} is called the closed
neighborhood of v. If uv and vw are distinct edges in G, then uv and vw are adjacent
edges. The vertex u and the edge uv are said to be incident with each other. Similarly,
v and uv are incident [7].

A graph of order 1 is called a trivial graph . A nontrivial graph therefore has two
or more vertices. A graph of size 0 is called an empty graph . A nonempty graph then
has one or more edges. In any empty graph, no two vertices are adjacent [7].

A u− v walk W in G is a sequence of vertices in G, beginning with u and ending at
v such that consecutive vertices in the sequence are adjacent. A u− v walk in a graph in
which no vertices are repeated is a u− v path. The distance dG(u, v) from a vertex u to
a vertex v in a connected graph G is the length of a shortest u− v path in G. If the graph



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 3 of 13

G being considered is understood, then this distance is written more simply as d(u, v). A
u− v path of length d(u, v) is called a u− v geodesic [7].

For an integer n ≥ 1, the path Pn is a graph of order n and size n− 1 whose vertices
can be labeled by v1, v2, ..., vn and whose edges are vivi+1 for i = 1, 2, ..., n− 1 [7].

For an integer n ≥ 3, the cycle Cn is a graph of order n and size n whose vertices can
be labeled by v1, v2, ..., vn and whose edges are v1vn and vivi+1 for i = 1, 2, ..., n− 1. The
cycle Cn is also referred to as an n−cycle [7] .

A complete graph of order n ≥ 2, denoted by Kn, is a graph with n vertices where
in every pair of distinct vertices are adjacent [7].

The friendship graph denoted by Frn is a set of n triangles having a common central
vertex [8].

A graph G is a complete bipartite graph denoted by Km,n if its vertices can be
partitioned into two disjoint nonempty sets V1 and V2 such that two vertices u and v are
adjacent if and only if u ∈ V1 and v ∈ V2. If |V1| = m and |V2| = n [9].

A star graph denoted by K1,n is a graph of order n+ 1 whose one vertex has degree
n which is called the apex u and the remaining n vertices have a degree equal to 1 [6].

For n ≥ 2, the fan graph Fn of order n+ 1 is a graph obtained by connecting a new
vertex v to each vertex of the path Pn [6].

A wheel graph Wn is a graph of order n + 1, where n ≥ 3 , which is obtained by
joining a new vertex called the root vertex of Wn to each of the vertices of the cycle Cn

produced from the complete product of an isolated vertex and a cycle Cn [6].

3. Results

This paper employs the term proximity prestige in social network analysis to represent
specific concepts in graph. Furthermore, for a graph G, the vertex set is denoted asV (G)
and the edge set as E(G), abbreviated to V and E, respectively.

Definition 1. Let G = (V,E) be a graph where V represents the set of vertices and E
represents the set of edges. The proximity prestige of a vertex vi ∈ V is defined as:

PPG(vi) =

∑
dG(vi, vj)

|V (G)|
where,

• PPG(vi) : proximity prestige of vertex vi;

• dG(vi, vj): length of the shortest path from vi to any vj; and

• |V (G)|: number of vertices in the graph.



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 4 of 13

Example 1. Consider the example below. By definition, if we choose v4, we have,

PPG(v4) =

∑
dG(v4, vj)

|V (G)|
=

1 + 1 + 1 + 2

5
=

5

5
= 1.

Figure 3.1: The Proximity Prestige of v4

v4

v1

v2

v3 v5

d(1)

d(2)

d(1)
d(1)

Special graph families considered in this paper are path Pn, cycle Cn, complete Kn,
friendship Frn, complete bipartite Km,n, star K1,n, fan Fn, and wheel Wn.

Theorem 1. Let G = (V,E) be a path graph Pn = [v1, v2, . . . vn] of order n ≥ 2, then the
proximity prestige of any vertex vi where 1 ≤ i ≤ n is given by,

PPP n(vi) =



(
n
2

)
n

, if i = 1 or i = n;

(i2 − i)

2n
+

(n− i)(n− i+ 1)

2n
, if 2 ≤ i ≤ n− i.

Proof. Considering the structure of path graph Pn = [v1, v2, . . . , vn] of order n ≥ 2,

PPP n(v1) = PPP n(vn) =
1 + 2 + 3 + . . .+ n− 1

n
=

(
n
2

)
n

. But for 2 ≤ i ≤ n− 1,

n−1∑
i=2

dP n(vi, vj) =

i−1∑
i=2

dP n(vi, vj) +

n−1∑
i=2

dP n(vi, vj)

= [1 + 2 + . . .+ (i− 1)] + [1 + 2 + . . .+ (n− i)]

=
(i2 − 1)

2
+

(n− i)(n− i+ 1)

2

Thus,

PPP n(vi) =
(i2 − 1)

2n
+

(n− i)(n− i+ 1)

2n
.



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 5 of 13

Therefore, we have

PPP n(vi) =



(
n
2

)
n

, if i = 1 or i = n;

(i2 − i)

2n
+

(n− i)(n− i+ 1)

2n
, if 2 ≤ i ≤ n− i.

■

Theorem 2. Let G = (V,E) be a cycle graph Cn = [v1, v2, . . . , vn, v1] of order n ≥ 3, then
the proximity prestige of any vertex vi where 1 ≤ i ≤ n is given by,

PPCn(vi) =


n

4
if n is even;

n2 − 1

4n
if n is odd.

Proof. Suppose first that n is even. By the structure of cycle Cn = [v1, v2, . . . , vn, v1],
the sum of the distance of vi and vj where i ̸= j can be derived as follows. For each i, we

have distance d(vi, vj) = 1 + 1 + 2 + 2 + . . .+ 2
(n
2
− 1

)
+

n

2
. Thus,

∑
i ̸=j

dCn(vi, vj) = 2
(
1 + 2 + . . .+

(n
2
− 1

))
+

n

2
=

n2

4
.

Hence,

PPCn(vi) =

n2

4
n

=
n

4
.

On the other hand, if n is odd, where i ̸= j. For each i, we have distance d(vi, vj) =

1 + 1 + 2 + 2 + . . .+ 2

(
n− 1

2

)
. Thus,

∑
i ̸=j

dCn(vi, vj) = 2

[
1 + 2 + . . .+

n− 1

2

]
=

n2 − 1

4
.

Hence,

PPCn(vi) =

n2 − 1

4
n

=
n2 − 1

4n
.



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 6 of 13

Therefore, we have

PPCn(vi) =


n

4
if n is even;

n2 − 1

4n
if n is odd.

■

Theorem 3. Let G be a complete graph Kn of order n ≥ 3, then the proximity prestige
of any vertex vi where 1 ≤ i ≤ n is given by,

PPKn(vi) =
n− 1

n
.

Proof. For each i, ∑
i ̸=j

dKn(vi, vj) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
n-1 addends

= n− 1.

Therefore,

PPKn(vi) =
n− 1

n
.

■

Theorem 4. Let G be a friendship graph Frn = [v1, v2, . . . , v2n,, v2n+1] where deg(v2n+1) =
2n , then the proximity prestige of any vertex vi where 1 ≤ i ≤ 2n+ 1 is given by,

PPFrn(vi) =


2n

2n+ 1
if deg(vi) = 2n;

4n− 2

2n+ 1
if deg(vi) = 2.

Proof. Consider the structure of a friendship graph which consists of n triangles sharing
a common vertex often called the center vertex v2n+1. Thus the total number of vertices
in Frn is 2n+ 1, with one center vertex of degree 2n and outer vertices each of degree 2.

Case 1: deg(vi) = 2n.
There is only one vertex with a degree of 2n, which is v2n+1 and for any i = 2n + 1 ̸= j
the distance d(vi, vj) = 1. Thus,∑

dFrn(vi, vj) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
2n addends

= 2n.

Hence,

PPFrn(vi) =
2n

2n+ 1
.



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 7 of 13

Case 2: deg (vi) = 2.
Choose v1 ∈ V (Frn), observe that the distance

d(v1, vj) = d(v1, v2n+1) + d(v1, v2) +
∑

j /∈{2,2n+1}

d(v1, vj).

Now, the distance d(v1, v2n+1) = 1 = d(v1, v2) and d(v1, vj) = 2 for j /∈ {2, 2n+ 1}. Thus,∑
j /∈{2,2n+1}

d(v1, vj) = 2 + 2 + . . .+ 2︸ ︷︷ ︸
2n-2 addends

= 2(2n− 2) = 4n− 4

Thus, ∑
dFrn(v1, vj) = 1 + 1 + 2 + 2 + . . .+ 2︸ ︷︷ ︸

2n-2 addends

= 2 + (4n− 4) = 4n− 2

Hence,

PPFrn(vi) =
4n− 2

2n+ 1
.

Therefore, we have

PPFrn(vi) =


2n

2n+ 1
if deg(vi) = 2n;

4n− 2

2n+ 1
if deg(vi) = 2.

■

Theorem 5. Let G = (V,E) be a fan graph Fn = [v1, . . . , vn, vn+1] where deg(vn+1) = n
, then the proximity prestige of any vertex vi where 1 ≤ i ≤ n+ 1 is given by,

PPF n(vi) =



n

n+ 1
, if deg(vi) = n;

2n− 2

n+ 1
, if deg(vi) = 2;

2n− 3

n+ 1
, if deg(vi) = 3.

Proof. Using the structure of fan graph Fn of order n ≥ 3, obtained by connecting
a single vertex vn+1 to each vertex of the path. Here, we need to consider three cases
separately.
Case 1: deg(vi) = n.



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 8 of 13

There is only one vertex with a degree n, which is vn+1 and for i = n+1 ̸= j the distance
d(vi, vj) = 1. Thus, ∑

i ̸=j

dF n(vi, vj) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
n addends

= n.

Hence,

PPF n(vi) =
n

n+ 1
.

Case 2: deg(vi) = 2.
If deg(vi) = 2, then there are only two vertices with a degree of 2 in Fn, that is v1 and vn
i.e., i ∈ {1, n}. Choose v1, thus the distance

d(v1, vj) = d(v1, v2) + d(v1, vn+1) +
∑

j /∈{2,n+1}

d(v1, vj).

Now, d(v1, v2) = 1 = d(v1, vn+1) and d(v1, vj) = 2 for j /∈ {2, n+ 1}. Thus,∑
j /∈{2,n+1}

d(v1, vj) = 2 + 2 + . . .+ 2︸ ︷︷ ︸
n-2 addends

= 2(n− 2) = 2n− 4.

Thus, ∑
dFn(v1, vj) = 1 + 1 + (2n− 4) = 2 + (2n− 4) = 2n− 2.

Hence,

PPF n(vi) =
2n− 2

n+ 1
.

Case 3: deg(vi) = 3.
Choose v2 ∈ V (Fn). Then the distance

d(v2, vj) = d(v2, v1) + d(v2, v3) + d(v2, vn+1) +
∑

j /∈{1,3,n+1}

d(v2, vj).

Now, d(v2, v1) = d(v2, v3) = d(v2, vn+1) = 1 and d(v2, vj) = 2, j /∈ {1, 3, n+ 1}. Thus,∑
j /∈{1,3,n+1}

d(v2, vj) = 2 + 2 + . . .+ 2︸ ︷︷ ︸
n-3 addends

= 2(n− 3) = 2n− 6

Thus, ∑
dF n(v2, vj) = 1 + 1 + 1 + 2 + 2 + . . .+ 2︸ ︷︷ ︸

n-3 addends

= 3 + (2n− 6) = 2n− 3.



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 9 of 13

Hence,

PPF n(vi) =
2n− 3

n+ 1
.

Therefore, we have

PPF n(vi) =



n

n+ 1
, if deg(vi) = n;

2n− 2

n+ 1
, if deg(vi) = 2;

2n− 3

n+ 1
, if deg(vi) = 3.

■

Theorem 6. Let G = (V,E) be a wheel graph Wn = [v1, v2, . . . , vn, vn+1] where deg(vn+1) =
n, then the proximity prestige of any vertex vi where 1 ≤ i ≤ n+ 1 is given by,

PPWn(vi) =


n

n+ 1
, if deg(vvi) = n;

2n− 3

n+ 1
, if deg(vi) = 3.

Proof. Using the structure of a wheel graph, formed by adjoining central vertex (vn+1)
to each vertex of the cycle Cn = [v1, v2, . . . , vn, v1] the following cases are need to be
considered.
Case 1: deg(vi) = n.
In this case, vi = vn+1, that is the central vertex of a wheel graph. There is only one
vertex with a degree n, that is vn+1 and distance d(vn+1, vj) = 1 for j ̸= n+ 1 since each
vertex {v1, v2, . . . , vn} is directly connected to the central vertex. Thus,∑

dWn(vi, vj) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
n addends

= n.

Hence,

PPWn(vi) =
n

2n+ 1
.

Case 2: deg(vi) = 3.
In this case, vi is any of the vertices {v1, v2, . . . , vn}. The distance from any vertex vi
where 1 ≤ i ≤ n to the central vertex vn+1 is 1 since each vi is adjacent to vn+1 and also
the distance from vi to its two adjacent vertices vi−1 and vi+1 in the cycle Cn is 1. Then



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 10 of 13

the distance from vi to any other vertex vj in the cycle where j /∈ {i, i− 1, i+ 1} is 2.
Thus the distance

d(vi, vj) = d(vi, vi−1) + d(vi, vi+1) + d(vi, vn+1) +
∑

j /∈{i−1,i+1,n+1}

d(vi, vj).

Now, from vi there are 3 vertices that have distance 1, that is d(vi, vi−1), d(vi, vi+1)and
d(vi, vn+1). Also, the remaining n− 3 vertices has distance 2 from vi. Thus,∑

dFn(vi, vj) = 1 + 1 + 1 + 2 + 2 + . . .+ 2︸ ︷︷ ︸
n-3 addends

= 3 + 2n− 6 = 2n− 3

Hence,

PPWn(vi) =
2n− 3

n+ 1
.

Therefore, we have

PPWn(vi) =


n

n+ 1
, if deg(vi) = n;

2n− 3

n+ 1
, if deg(vi) = 3.

■

Theorem 7. Let G = (V,E) be a complete bipartite Km,n, then the proximity prestige of
any vertex vi where 1 ≤ i ≤ m+ n is given by,

PPKm,n(vi) =


n+ (2m− 2)

m+ n
, if deg(vi) = n;

m+ (2n− 2)

m+ n
, if deg(vi) = m.

Proof. Using the structure of complete bipartite graph, formed if its vertices can be
partitioned into two disjoint nonempty sets V1 and V2 such that two vertices u and v
are adjacent if and only if u ∈ V1 and v ∈ V2 then the following cases are needed to be
consider.
Case 1: deg(vi) = n.
Then, ∑

i ̸=j

dKm,n(vi, vj) =
∑

1≤q≤n

d(vi, vq) +
∑

k ̸=i,1≤k≤m

d(vi, vk).



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 11 of 13

Now, ∑
1≤q≤n

d(vi, vq) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
n addends

= n

and ∑
k ̸=i,1≤k≤m

d(vi, vk) = 2 + 2 + . . .+ 2︸ ︷︷ ︸
m-1 addends

= 2(m− 1) = 2m− 2.

Thus, ∑
i ̸=j

dKm,n(vi, vj) = n+ 2m− 2.

Hence,

PPKm,n(vm) =
n+ 2m− 2

m+ n
.

Case 2: deg(vi) = m.
Then, ∑

i ̸=j

dKm,n(vi, vj) =
∑

1≤k≤m

d(vi, vk) +
∑

q ̸=i,1≤q≤n

d(vi, vq).

Now,

∑
1≤k≤m

d(vi, vk) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
m addends

= m

and ∑
l ̸=i,1≤q≤n

d(vi, vq) = 2 + 2 + . . .+ 2︸ ︷︷ ︸
n-1 addends

= 2(n− 1) = 2n− 2.

Thus, ∑
i ̸=j

dKm,n(vi, vj) = m+ (2n− 2).



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 12 of 13

Hence,

PPKm,n(vn) =
m+ (2n− 2)

m+ n
.

Therefore, we have

PPKm,n(vi) =


n+ (2m− 2)

m+ n
, if deg(vi) = n;

m+ (2n− 2)

m+ n
, if deg(vi) = m.

■

Theorem 8. Let G = (V,E) be a star K1,n = [v1, v2, . . . , vn, vn+1] where deg(vn+1) = n,
then the proximity prestige of any vertex vi where 1 ≤ i ≤ n+ 1 is given by,

PP (vi) =


n

n+ 1
, if deg(vi) = n;

2n− 1

n+ 1
, if deg(vi) = 1.

Proof. Suppose first that deg(vi) = n. There is only one vertex with a degree n in
K1,n, that is vn+1, and the distance d(vi, vj) = 1 , for j ̸= n+ 1. Thus,∑

j ̸=n+1

dK1,n(vi, vj) = 1 + 1 + . . .+ 1︸ ︷︷ ︸
n addends

= n.

Hence,

PPK1,n(vi) =
n

n+ 1
.

On the other hand, if deg(vi) = 1. Then,∑
i ̸=j

dK1,n(vi, vj) = 1 + 2 + 2 + . . .+ 2︸ ︷︷ ︸
n-1 addends

= 1 + 2(n− 1) = 2n− 1

Hence,

PPK1,n(vi) =
2n− 1

n+ 1
.

Therefore, we have

PPK1,n(vi) =


n

n+ 1
, if deg(vi) = n;

2n− 1

n+ 1
, if deg(vi) = 1.

■



L. Toladro, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5943 13 of 13

4. Conclusion

This paper introduced proximity prestige (PP ) as a centrality measure in fixed graphs,
defined by the average shortest path distance from a vertex to all other vertices, focusing on
indirect connections. Proximity prestige (PP ) offers a valuable approach for quantifying
vertex importance based on its reach within the network.

Future research could explore the application of proximity prestige (PP ) to random
and dynamic graphs, as well as its integration with other centrality measures, to enhance
understanding of vertex influence in evolving network structures and real-world, complex
networks.

Acknowledgements

The authors would like to express their sincere thanks to everyone who contributed
to the successful completion of this research. In particular, they gratefully acknowledge
the invaluable support provided by the Department of Science and Technology-Science
Education Institute Science and Technology Regional Alliance of Universities for Inclusive
National Development (DOST-SEI STRAND) throughout the study. The authors also
wish to extend their heartfelt appreciation to the referees for their constructive feedback
and insightful suggestions, which greatly enhanced the quality of this work.

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