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Graph Labelings: A Prime Area to Explore

• First Online: 05 July 2022

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• Elizabeth A. Donovan 5 &
• Lesley W. Wiglesworth 6  

Part of the book series: Foundations for Undergraduate Research in Mathematics ((FURM))

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A graph labeling assigns integers to the vertices or edges (or both) of a graph, subject to certain conditions. Labelings were first studied in the late 1960s, making them a relatively new area of mathematics to explore. There are hundreds of different types of labelings, many of which are motivated by circuit design, coding theory, and communication networks. In this chapter, we introduce the reader to prime labelings, a labeling of the vertices in such a way that the labels of adjacent vertices are relatively prime. We also introduce several variations of prime labelings while guiding the reader through examples and existing research. We present the reader with several research questions for each variation, several of which have never been explored, thus allowing the reader to make progress on new problems.

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Department of Mathematics and Statistics, Murray State University, Murray, KY, USA

Elizabeth A. Donovan

Department of Mathematics, Centre College, Danville, KY, USA

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Donovan, E.A., Wiglesworth, L.W. (2022). Graph Labelings: A Prime Area to Explore. In: Goldwyn, E.E., Ganzell, S., Wootton, A. (eds) Mathematics Research for the Beginning Student, Volume 1. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-08560-4_4

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Graph Labeling

• Joseph A. Gallian

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid 1960s. In the intervening 50 years nearly 200 graph labelings techniques have been studied in over 2000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey, I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index. This edition has 55 additional pages and 308 new references that are identified with the reference number and the word “new” in the right margin.

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• Published: 09 July 2019

Edge even graceful labeling of some graphs

• Mohamed R. Zeen El Deen   ORCID: orcid.org/0000-0002-1100-3309 1  

Journal of the Egyptian Mathematical Society volume  27 , Article number:  20 ( 2019 ) Cite this article

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Edge even graceful labeling is a new type of labeling since it was introduced in 2017 by Elsonbaty and Daoud (Ars Combinatoria 130:79–96, 2017). In this paper, we obtained an edge even graceful labeling for some path-related graphs like Y- tree, the double star B n , m , the graph 〈 K 1,2 n : K 1,2 m 〉, the graph \( ~P_{2n-1}\odot \overline { K_{2m}}~ \) , and double fan graph F 2, n . Also, we showed that some cycle-related graphs like the prism graph \( ~\prod _{n}~ \) , the graph \( ~C_{n}\left (\frac {n}{2}\right)~ \) , the flag FL n , the graph K 2 ⊙ C n , the flower graph FL( n ), and the double cycle { C n , n } are edge even graphs.

Introduction

A graph labeling is an assignment of integers to the edges or vertices, or both, subject to certain condition. The idea of graph labelings was introduced by Rosa in [ 1 ]. Following this paper, other studies on different types of labelings (Odd graceful, Chordal graceful, Harmonious, edge odd graceful) introduced by many others [ 2 – 4 ]. A new type of labeling of a graph called an edge even graceful labeling has been introduced by Elsonbaty and Daoud [ 5 ]. They introduced some path- and cycle-related graphs which are edge even graceful.

Graph labelings give us useful models for a wide range of applications such as coding theory, X-ray, astronomy, radar, and communication network addressing.

Definition 1

[ 5 ] An edge even graceful labeling of a graph G ( V ( G ), E ( G )) with p =| V ( G )| vertices and q =| E ( G )| edges is a bijective mapping f of the edge set E ( G ) into the set {2,4,6, ⋯ ,2 q } such that the induced mapping f ∗ : V ( G ) →{0,2,4, ⋯ ,2 q }, given by: \( f^{\ast }(x) = \left ({\sum \nolimits }_{xy \in E(G)} f(xy)~ \right)~mod~(2k) \) , is an injective function, where k = m a x ( p , q ). The graph that admits an edge even graceful labeling is called an edge even graceful graph.

In Fig.  1 , we present an edge even graceful labeling of the Peterson graph and the complete graph K 5 .

An edge even graceful labeling of Peterson graph and the complete graph K 5

Edge even graceful for some path related graphs

A Y- tree is a graph obtained from a path by appending an edge to a vertex of a path adjacent to an end point, and it is denoted by Y n where n is the number of vertices in the tree.

The Y-tree Y n is an edge even graceful graph when n is odd.

The number of vertices of Y-tree Y n is n and the number of edges is n −1. Let the vertices and the edges of Y n be given as in Fig.  2 .

Y n with ordinary labeling

We define the mapping f : E ( Y n ) →{2,4, ⋯ ,2 n −2} as follows:

Then, the induced vertex labels are

Similarly, \( ~~~f^{\ast }\left (v_{\frac {n+3}{2}}\right)= 6~,~~~~~~~~~~~~~ f^{\ast }\left (v_{\frac {n+5}{2}}\right)= 8~, f^{\ast }\left (v_{\frac {n+7}{2}}\right)= 10 ~\) ,

Clearly, all the vertex labels are even and distinct. Thus, Y-tree Y n is an edge even graceful graph when n is odd. □

Illustration: The edge even labeling of the graph Y 13 is shown in Fig.  3 .

The edge even labeling of the graph Y 13

Double star is the graph obtained by joining the center of the two stars K 1, n and K 1, m with an edge, denoted by B n , m . The graph B n , m has p = n + m +2 and q = n + m +1.

The double star B n , m is edge even graceful graph when one ( m o r n ) is an odd number and the other is an even number.

Without loss of generality, assume that n is odd and m is even. Let the vertex and edge symbols be given as in Fig.  4 .

B m , n with ordinary labeling

Define the mapping f : E ( B m , n ) →{2,4, ⋯ ,2 q } as follows:

We realize the following:

So, the vertex labels will be

Also, each pendant vertex takes the labels of its incident edge which are different from the labels of the vertices U and v . □

The graph 〈 K 1, n : K 1, m 〉 is obtained by joining the center v 1 of the star K 1, n and the center v 2 of the another star K 1, m to a new vertex u , so the number of vertices is p = n + m +3 and the number of edges is q = n + m +2.

The graph 〈 K 1,2 n : K 1,2 m 〉 is an edge even graceful graph.

Let the vertex and edge symbols be given as in Fig.  5 .

〈 K 1,2 n : K 1,2 m 〉 with ordinary labeling

Define the mapping f : E ( G ) →{2,4, ⋯ ,2 p } as follows:

We can see the following:

which are even and distinct from each pendant vertices. Thus, the graph 〈 K 1,2 n : K 1,2 m 〉 is an edge even graceful graph. □

The corona [ 6 ] \( ~P_{3}\odot \overline { K_{2m}}~ \) is an edge even graceful graph.

In this graph p =6 m +3 and q =6 m +2. Let the vertex and edge symbols be given as in Fig.  6 .

\( ~P_{3}\odot \overline { K_{2m}}~ \) with ordinary labeling

We can define the mapping \(f: E(P_{3}\odot \overline { K_{2m}}~)~\rightarrow \{2,4,\cdots, 2q \}\) as follows:

It is clear that f ( A ij )+ f ( B ij )≡0 mod (2 p ) for j =1,2, ⋯ , m

So the vertex labels will be,

Therefore, all the vertices are even and distinct which complete the proof. □

Illustration: The edge even labeling of the graph \( ~P_{3}\odot \overline { K_{6}}~ \) is shown in Fig.  7 .

Edge even graceful labeling of the graph \( ~P_{3}\odot \overline { K_{6}}~ \)

The generalization of the previous result is presented in the following theory

The graph \( ~P_{2n-1}\odot \overline { K_{2m}}~ \) is an edge even graceful graph.

In this graph, p =4 n m +2( n − m )−1 and q =4 n m +2( n − m )−2. The middle vertex in the path P 2 n −1 will be v n , and we start the labeling from this vertex. Let the vertex and edge symbols be given as in Fig.  8 .

\( ~P_{2n-1}\odot \overline { K_{2m}}~ \) with ordinary labeling

Define the mapping f : E ( G ) →{2,4, ⋯ ,2 q } by the following arrangement

f ( e i )=2 i for i =1,2, ⋯ , n −1

f ( b i )=2 p − 2 i for i =1,2, ⋯ , n −1

f ( A ij ) will take any number from the reminder set of the labeling not contains 2 i nor 2 p − 2 i for j =1,2, ⋯ , m and

f ( D ij )=2 p −[ f ( A ij )] for j =1,2, ⋯ , m .

It is clear that [ f ( A ij )+ f ( D ij )] mod (2 p )≡0 mod (2 p ) for j =1,2, ⋯ , m .

f ∗ ( v n )=[ f ( e 1 )+ f ( b 1 )] mod (2 p ) ≡0 mod (2 p ),

For any vertex v k when k < n , let k = n − i and i =1,2, ⋯ , n −2

When k > n , let k = n + i and i =1,2, ⋯ , n −2

The pendant vertices v 1 a n d v 2 n −1 of the path P 2 n −1 will take the labels of its pendant edges of P 2 n −1 ,i.e.,

If n is even, then f ∗ ( v 1 )= f ( e n −1 ), and f ∗ ( v 2 n −1 )= f ( b n −1 )

If n is odd, then f ∗ ( v 1 )= f ( b n −1 ), and f ∗ ( v 2 n −1 )= f ( e n −1 )

Then, the labels of the vertices of the path P 2 n −1 takes the labels of the edges of the path P 2 n −1 , and each pendant vertex takes the labels of its incident edge. Then, there are no repeated vertex labels, which complete the proof. □

Illustration: The graph \( ~P_{11}\odot \overline { K_{4}}~ \) labeled according to Theorem 1 is presented in Fig.  9 .

Edge even graceful labeling of the graph \( ~P_{11}\odot \overline { K_{4}}~ \)

A double fan graph F 2, n is defined as the graph join \(\overline {K_{2}} + P_{n}~\) where \(\overline {K_{2}} \) is the empty graph on two vertices and P n be a path of length n .

The double fan graph F 2, n is an edge even graceful labeling when n is even.

In the graph F 2, n we have p = n +2 a n d q =3 n −1. Let the graph F 2, n be given as indicated in Fig.  10 .

F 2, n with ordinary labeling

Define the edge labeling function f : E ( F 2, n )→ {2,4, ⋯ , 6 n −2} as follows:

f ( a i )=2 i ; i =1,2, ⋯ n

f ( b i )=2 q −2 i =6 n −2( i +1); i =1,2, ⋯ n

Hence, the induced vertex labels are

\(~ f^{\ast }(u) = \left (\sum _{i=1}^{n}(f(a_{i}))\right)~\text {mod}(6n-2)\,=\, \left (\sum _{i=1}^{n}(2i)\right)~\text {mod}(6n-2) =(n^{2}+n)~\text {mod}(6n-2)~~\)

\(~ f^{\ast }(v) = \left (\sum _{i=1}^{n}(f(b_{i})) \right)~\text {mod}(6n-2)= \left (\sum _{i=1}^{n}(2q-2i)\right)~\text {mod}(6n-2) =\left (- n^{2}-n\right)~\text {mod}(6n-2) \)

\( f^{\ast }(v_{i}) \,=\, \left [\sum _{i=1}^{n}(f(a_{i})\,+\,f(b_{i})\,+\,f (e_{i}) \,+\,f(e_{i-1}))\right ]~\text {mod}(6n\,-\,2)~ \,=\,(4n\,+\,4i\,-\,2)~\text {mod}(6n\,-\,2)~,~~2 \leq i \leq \frac {n}{2}- 1 \)

\(f^{\ast }(v_{i}) = \left [\sum _{i=1}^{n}(f(a_{i})+f(b_{i})+f (e_{i}) +f(e_{i-1}))\right ]~\text {mod}(6n-2) =(4n+4i-6)~\text {mod}(6n-2),~~\frac {n}{2}+2 \leq i \leq n- 1 \)

Since [ f ( a i )+ f ( b i )] mod(6 n −2)=0, we see that

f ∗ ( v 1 )= f ( e 1 )=2 n +2,

f ∗ ( v n )= f ( e n −1 )=4 n −4,

\(~~~~~ f^{\ast }(v_{\frac {n}{2}})=~f (e_{\frac {n}{2}-1})= 3n-2 ~\) and

\( ~~~~f^{\ast }(v_{\frac {n}{2}+1})=~f (e_{\frac {n}{2}})= 3n ~\)

Thus, the set of vertices \( v_{1}, v_{2},v_{3},\cdots ~,v_{\frac {n}{2}-1},v_{\frac {n}{2} },v_{\frac {n}{2}+1},v_{\frac {n}{2}+2},v_{\frac {n}{2}+3},\cdots ~,v_{n-2},v_{n-1},v_{n} \) are labeled by 2 n +2, 4 n +6, 4 n +10, ⋯ ,6 n −6, 3 n −2, 3 n , 4,8, ⋯ ,2 n −12, 2 n −8, 4 n −4 respectively.

Clearly, f ∗ ( u ) and f ∗ ( v ) are different from all the labels of the vertices. Hence F 2, n is an edge even graceful when n is even. □

Illustration: The double fan F 2,10 labeled according to Theorem 2 is presented in Fig.  11 .

Edge even graceful labeling of the double fan F 2,10

Edge even graceful for some cycle related graphs

Definition 2.

For n ≥4, a cycle (of order n) with one chord is a simple graph obtained from an n -cycle by adding a chord. Let the n -cycle be v 1 v 2 ⋯ v n v 1 . Without loss of generality, we assume that the chord joins v 1 with any one v i , where 3≤ i ≤ n −1. This graph is denoted by C n ( i ).

The graph \( ~C_{n}\left (\frac {n}{2}\right)~ \) is an edge even graceful graph if n is even.

Let \( \{ v_{1}, v_{2},\cdots, v_{\frac {n}{2}-1},v_{\frac {n}{2}},v_{\frac {n}{2}+1}, \cdots,v_{n} \}\) be the vertices of the graph \(C_{n}\left (\frac {n}{2}\right)\) , and the edges are e i =( v i v i +1 ) for i ≤ i ≤ n −1 and the chord \(e_{0}=(v_{1}v_{\frac {n}{2}}) \) connecting the vertex v 1 with \(v_{\frac {n}{2}}\) as in Fig.  12 .

\( ~C_{n}\left (\frac {n}{2}\right)~ \) with ordinary labeling

Here, p = n and q = n +1, so 2 k =2 q =2 n +2; first, we label the edges as follows:

Then, the induced vertex labels are as follows:

for any other vertex \(~~~v_{i}, ~~~i\neq 1,\frac {n}{2}\)

Hence, the labels of the vertices \(v_{0}, v_{1},v_{2}, \cdots,v_{\frac {n}{2}-1},v_{\frac {n}{2}},v_{\frac {n}{2}+1}, \cdots v_{n} \) are 6, 10, 14, ⋯ , 2 n −2, 2, 4, ⋯ , 2 n respectively, which are even and distinct. So, the graph \( ~C_{n}\left (\frac {n}{2}\right)~ \) is an edge even graceful graph if n is even. □

Definition 3

Let C n denote the cycle of length n . The flag F L n is obtained by joining one vertex of C n to an extra vertex called the root, in this graph p = q = n +1.

The flag graph F L n is edge even graceful graph when n is even.

Let { v 1 , v 2 , ⋯ , v n } be the vertices of the cycle C n and the edges are e i =( v i v i +1 ) f o r 1≤ i ≤ n and the edge e =( v 1 v 0 ) connecting the vertex v 1 with v 0 as in Fig.  13 .

F l n with ordinary labeling

First, we label the edges as follows:

Then, the induced vertex labels are as follows f ∗ ( v 0 )= f ( e 0 )=2,

Hence, the labels of the vertices \(v_{0}, v_{1},v_{2}, \cdots,v_{\frac {n}{2}-1},v_{\frac {n}{2}},v_{\frac {n}{2}+1}, \cdots v_{n} \) will be 2, 6, 10, 14, ⋯ , 2 n −2, 0, 4, ⋯ , 2 n respectively. □

The graph K 2 ⊙ C n is edge even graceful graph when n is odd.

Let \( \left \{ v_{1}, v_{2},\cdots, v_{n},v^{'}_{1},v^{'}_{2}, \cdots,v^{'}_{n} \right \}\) be the vertices of the graph K 2 ⊙ C n and the edges are \( \left \{e_{1}, e_{2},\cdots, e_{n},~e^{'}_{1},e^{'}_{2}, \cdots,e^{'}_{n} \right \}\) as shown in Fig.  14 . Here, p =2 n and q =2 n +1, so 2 k =4 n +2.

The graphs K 2 ⊙ C n with ordinary labeling

We can see that [ f ( e n )+ f ( e n +1 )]mod(2 q )=(4 n +2) mod(4 n +2)≡0

Then, the induced vertex labels are as follows

\( f^{\ast }(v^{\prime }_{i})= \left [ f(e^{\prime }_{i}) +f(e^{\prime }_{i+1})\right ]~\text {mod}(4n+2)=[8+4~(i-1)]~\text {mod}(4n+2)~,~~1 \leq i < n \)

f ∗ ( v n )= [ f ( e 1 )+ f ( e n )+ f ( e n +1 ) ] mod(4 n +2)= f ( e 1 ) mod(4 n +2) ≡2

\( ~~~f^{\ast }(v^{\prime }_{n})=~ \left [ f(e^{\prime }_{1}) +f(e_{n+1})~+~f\left (e^{\prime }_{n}\right)\right ]~\text {mod}(4n+2) ~\equiv ~4\)

Clearly, the vertex labels are all even and distinct. Hence, the graph K 2 ⊙ C n is edge even graceful for odd n . □

Let C n denote the cycle of length n . Then, the corona of all vertices of C n except one vertex { v 1 } with the complement graph \(\overline {K_{2m-1}}\) is denoted by \( ~ \{C_{n}-\{v_{1} \} \} \odot \overline {K_{2m-1}}~ \) , in this graph p = q =2 m ( n −1)+1.

The graph \( \{C_{n}-\{v_{1} \} \} \odot \overline {K_{2m-1}}~ \) is an edge even graceful graph.

Let the vertex and edge symbols be given as in Fig.  15 .

\( ~ \{C_{n}-\{v_{1} \} \} \odot \overline {K_{2m-1}}~ \) with ordinary labeling

Define the mapping f : E ( G ) →{2,4, ⋯ ,2 q } as follows:

[ f ( A ij )+ f ( B ij )] mod (2 q )≡0 mod (2 q ) for j =1,2, ⋯ , m −1

Also, [ f ( E i −1 )+ f ( e i )] mod (2 q )≡0 mod (2 q ) f o r i =2,3, ⋯ , n

So, verifying the vertex labels, we get that,

f ∗ ( v 1 )=[ f ( E 1 )+ f ( E n )] mod (2 q )=(2+2 q ) mod (2 q )=2,

Hence, the labels of the vertices v 1 , v 2 , ⋯ , v n takes the label of the edges of the cycles and each of the pendant vertices takes the label of its edge, so they are all even and different numbers. □

Illustration: In Fig.  16 , we present an edge even graceful labeling of the graph \( ~ \{C_{6}-\{v_{1} \} \} \odot \overline {K_{3}}~ \) .

\( ~ \{C_{6}-\{v_{1} \} \} \odot \overline {K_{3}}~ \)

The double cycle graph { C n , n } is an edge even graceful graph when n is odd.

Here, p = n and q =2 n . Let the vertex and edge symbols be given as in Fig.  17 .

{ C n , n } with ordinary labeling

Define the mapping f : E ( G ) →{2,4, ⋯ ,4 n } by

f ( e i )=2 i for i =1,2, ⋯ , n . So, the vertex labels will be

f ∗ ( v 1 )=[ f ( e 1 )+ f ( e n ) + f ( e n +1 ) + f ( e 2 n )]mod (4 n )=4

Hence, the labels of the vertices \( v_{1},v_{2},v_{3}, \cdots, v_{\frac {n+1}{2}}, v_{\frac {n+3}{2}}, \cdots, v_{n} \) will be 4,12,20, ⋯ ,0,8, ⋯ ,4 n −4 □

The prism graph \( \prod _{n}\) is the cartesian product C n □ K 2 of a cycle C n by an edge K 2 , and an n -prism graph has p =2 n vertices and q =3 n edges.

The prism graph \( ~\prod _{n}~ \) is edge even graceful graph.

In the prism graph \( \prod _{n}\) we have two copies of the cycle C n , let the vertices in one copy be v 1 , v 2 , ⋯ , v n and the vertices on the other copy be \(~v^{\prime }_{1}, v^{\prime }_{2}, \cdots, v^{\prime }_{n}~\) . In \( ~\prod _{n}~ \) , the edges will be

\(v_{i}v_{i+1}, ~~ ~~~v^{\prime }_{i}v^{\prime }_{i+1},~~~ \text {and} ~~~v_{i}v^{\prime }_{i} \) . Let the vertex and edge symbols be given as in Fig.  18 .

\( ~\prod _{n}~ \) with ordinary labeling

Define the mapping \(f: E\left (\prod _{n}\right)~\rightarrow \{2,4,\cdots, 6n \}\) by

Hence, the labels of the vertices v 1 , v 2 , ⋯ , v n will be 4 n +4, 4 n +6, ⋯ ,0, 2 respectively.

Also, \( f^{\ast }(v^{\prime }_{1}) = \left [~ f(e^{\prime }_{1}) +f(e^{\prime }_{n})~+ f(E_{1})\right ] \text {mod}~(6n)= 12n+ 4~ \text {mod}~(6n)= 4\)

Hence, the labels of the vertices \( v^{\prime }_{1},v^{\prime }_{2}, \cdots, v^{\prime }_{n} \) are 4, 6,8, ⋯ ,2 n , 2 n +2 respectively. Overall, the vertices are even and different. Thus, the prism graph \( ~\prod _{n}~ \) is an edge even graceful graph. □

Illustration: In Fig.  19 , we present an edge even graceful labeling of of prism graphs \( ~\prod _{5}~ \) and \( ~\prod _{6} \) .

An edge even graceful labeling of prism graphs \( ~\prod _{5}~ \) and \( ~\prod _{6} \)

The flower graph FL( n ) ( n ≥3) is the graph obtained from a helm H n by joining each pendant vertex to the center of the helm.

The flower graph FL(n) ( n ≥4) is an edge even graceful graph.

In the flower graph FL( n ) ( n ≥4), we have p =2 n +1 and q =4 n . Let \(\{~v_{0},v_{1}, v_{2}, \cdots, v_{n}~,v^{\prime }_{1}, v^{\prime }_{2}, \cdots, v^{\prime }_{n}\}\) be the vertices of FL( n ) and

{ e 1 , e 2 , e 3 , ⋯ , e 3 n , E 1 , E 2 , E 3 , ⋯ , E n } be the edges of FL( n ) as in Fig.  20 .

The flower graph FL(n) with ordinary labeling

First, define the mapping f : E ( F l ( n ))→{2,4, ⋯ ,8 n } as the following:

Overall, all the vertex labels are even and distinct which complete the proof. □

Illustration: In Fig.  21 , we present an edge even graceful labeling of of the flower graphs FL(6)and FL(7).

An edge even graceful labeling of The flower graphs FL(6) and FL(7)

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Zeen El Deen, M.R. Edge even graceful labeling of some graphs. J Egypt Math Soc 27 , 20 (2019). https://doi.org/10.1186/s42787-019-0025-x

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<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

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A method for relaxed graceful labeling of P2n graphs is presented together with an algorithm designed for labeling these graphs. Graceful labeling is achieved by relaxing the range to 2m and perform the labeling using an algorithm with quadratic complexity (O(n2)). The algorithm can be used for labeling any P2n graph with n ≥ 3, as far as the machine can handle the size of the problem.

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Edge even graceful labeling of some corona graphs

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A Dual-channel Progressive Graph Convolutional Network via subgraph sampling

• Wenrui Guan 1,2 , 
• Xun Wang 1 ,  , 
• 1. School of Computer, Jiangsu University of Science and Technology, Zhenjiang 212100, China
• 2. School of Automation, Jiangsu University of Science and Technology, Zhenjiang 212100, China
• Received: 12 May 2024 Revised: 25 June 2024 Accepted: 01 July 2024 Published: 11 July 2024
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Graph Convolutional Networks (GCNs) demonstrate an excellent performance in node classification tasks by updating node representation via aggregating information from the neighbor nodes. Note that the complex interactions among all the nodes can produce challenges for GCNs. Independent subgraph sampling effectively limits the neighbor aggregation in convolutional computations, and it has become a popular method to improve the efficiency of training GCNs. However, there are still some improvements in the existing subgraph sampling strategies: 1) a loss of the model performance caused by ignoring the connection information among different subgraphs; and 2) a lack of representation power caused by an incomplete topology. Therefore, we propose a novel model called Dual-channel Progressive Graph Convolutional Network (DPGCN) via sub-graph sampling. We construct subgraphs via clustering and maintain the connection information among the different subgraphs. To enhance the representation power, we construct a dual channel fusion module by using both the geometric information of the node feature and the original topology. Specifically, we evaluate the complementary information of the dual channels based on the joint entropy between the feature information and the adjacency matrix, and effectively reduce the information redundancy by reasonably selecting the feature information. Then, the model convergence is accelerated through parameter sharing and weight updating in progressive training. We select 4 real datasets and 8 characteristic models for comparison on the semi-supervised node classification task. The results verify that the DPGCN possesses superior classification accuracy and robustness. In addition, the proposed architecture performs excellently in the low labeling rate, which is of practical value to label scarcity problems in real cases.

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Citation: Wenrui Guan, Xun Wang. A Dual-channel Progressive Graph Convolutional Network via subgraph sampling[J]. Electronic Research Archive, 2024, 32(7): 4398-4415. doi: 10.3934/era.2024198

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Abstract: In pharmaceutical research, the strategy of drug repurposing accelerates the development of new therapies while reducing R&D costs. Network pharmacology lays the theoretical groundwork for identifying new drug indications, and deep graph models have become essential for their precision in mapping complex biological networks. Our study introduces an advanced graph model that utilizes graph convolutional networks and tensor decomposition to effectively predict signed chemical-gene interactions. This model demonstrates superior predictive performance, especially in handling the polar relations in biological networks. Our research opens new avenues for drug discovery and repurposing, especially in understanding the mechanism of actions of drugs.

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Analysis of agrometeorological hazard based on knowledge graph.

1. Background

2. research methods, 2.1. knowledge graph construction, 2.2. text recognition, 2.3. knowledge verification, 3. construction of knowledge graph for agricultural meteorological hazard analysis, 3.1. hazard ontology modeling, 3.1.1. model definition, 3.1.2. instantiation, 3.2. knowledge extraction, 3.2.1. agricultural production data extraction, 3.2.2. meteorological hazard data extraction, 3.3. knowledge fusion, 3.4. knowledge storage, 4. analysis template for agrometeorological hazards, 4.1. identification of hazard related entities, 4.2. hazard analysis template, 5. result analysis, 5.1. graph construction, 5.1.1. knowledge graph entity relationships, 5.1.2. results of knowledge graph, 5.2. hazard ontology modeling, 5.3. hazard analysis results, 6. conclusions and outlook, author contributions, institutional review board statement, data availability statement, conflicts of interest.

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Wu, D.; Liu, X.; Zai, S.; Zhang, L.; Feng, X. Analysis of Agrometeorological Hazard Based on Knowledge Graph. Agriculture 2024 , 14 , 1130. https://doi.org/10.3390/agriculture14071130

Wu D, Liu X, Zai S, Zhang L, Feng X. Analysis of Agrometeorological Hazard Based on Knowledge Graph. Agriculture . 2024; 14(7):1130. https://doi.org/10.3390/agriculture14071130

Wu, Di, Xuemei Liu, Songmei Zai, Liang Zhang, and Xuefang Feng. 2024. "Analysis of Agrometeorological Hazard Based on Knowledge Graph" Agriculture 14, no. 7: 1130. https://doi.org/10.3390/agriculture14071130

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Towards Complete Causal Explanation with Expert Knowledge

Published 7/9/2024

Flawed Autopsy ‘Review’ Revives Unsupported Claims of COVID-19 Vaccine Harm, Censorship

By Jessica McDonald

Posted on July 5, 2024

SciCheck Digest

COVID-19 vaccination is generally very safe, and except for extremely rare cases, there is no evidence that it contributes to death. Social media posts about a now-published, but faulty review of autopsy reports, however, are repeating an unfounded claim from last summer that “74% of sudden deaths are shown to be due to the COVID-19 vaccine.”

More than  half a billion doses of COVID-19 vaccines have now been administered in the U.S. and only a few, very rare, safety concerns have emerged. The vast majority of people experience only minor, temporary side effects such as pain at the injection site, fatigue, headache, or muscle pain — or no side effects at all. As the Centers for Disease Control and Prevention has said , these vaccines “have undergone and will continue to undergo the most intensive safety monitoring in U.S. history.”

A small number of severe allergic reactions known as anaphylaxis, which are expected with any vaccine, have occurred with the authorized and approved COVID-19 vaccines. Fortunately, these reactions are rare, typically occur within minutes of inoculation and can be treated. Approximately 5 per million people vaccinated have experienced anaphylaxis after a COVID-19 vaccine, according  to the CDC.

To make sure serious allergic reactions can be identified and treated, all people receiving a vaccine should be observed for 15 minutes after getting a shot, and anyone who has experienced anaphylaxis or had any kind of immediate allergic reaction to any vaccine or injection in the past should be monitored for a half hour. People who have had a serious allergic reaction to a previous dose or one of the vaccine ingredients should not be immunized. Also, those who shouldn’t receive one type of COVID-19 vaccine should be monitored for 30 minutes after receiving a different type of vaccine.

There is evidence that the Pfizer/BioNTech and Moderna mRNA vaccines may rarely cause inflammation of the heart muscle (myocarditis) or of the surrounding lining (pericarditis), particularly in male adolescents and young adults .

Based on data collected through August 2021, the reporting rates of either condition in the U.S. are highest in males 16 to 17 years old after the second dose (105.9 cases per million doses of the Pfizer/BioNTech vaccine), followed by 12- to 15-year-old males (70.7 cases per million). The rate for 18- to 24-year-old males was 52.4 cases and 56.3 cases per million doses of Pfizer/BioNTech and Moderna vaccines, respectively.

Health officials have emphasized that vaccine-related myocarditis and pericarditis cases are rare and the benefits of vaccination still outweigh the risks. Early evidence suggests these myocarditis cases are less severe than typical ones. The CDC has also noted that most patients who were treated “responded well to medicine and rest and felt better quickly.”

The Johnson & Johnson vaccine has been linked to an  increased risk of rare blood clots combined with low levels of blood platelets, especially in women ages 30 to 49 . Early symptoms of the condition, which is known as thrombosis with thrombocytopenia syndrome, or TTS, can appear as late as three weeks after vaccination and  include  severe or persistent headaches or blurred vision, leg swelling, and easy bruising or tiny blood spots under the skin outside of the injection site.

According to the CDC, TTS has occurred in around 4 people per million doses administered. As of early April ,  the syndrome has been confirmed in 60 cases, including nine deaths, after more than 18.6 million doses of the J&J vaccine. Although TTS remains rare, because of the availability of mRNA vaccines, which are not associated with this serious side effect, the FDA on May 5 limited authorized use of the J&J vaccine to adults who either couldn’t get one of the other authorized or approved COVID-19 vaccines because of medical or access reasons, or only wanted a J&J vaccine for protection against the disease. Several months earlier, on Dec. 16, 2021 ,  the CDC had recommended the Pfizer/BioNTech and Moderna shots over J&J’s.

The J&J vaccine has also been linked to an increased risk of Guillain-Barré Syndrome, a rare disorder in which the immune system attacks nerve cells.  Most people  who develop GBS fully recover, although some have permanent nerve damage and the condition can be fatal.

Safety surveillance data suggest that compared with the mRNA vaccines, which have not been linked to GBS, the J&J vaccine is associated with 15.5 additional GBS cases per million doses of vaccine in the three weeks following vaccination. Most reported cases following J&J vaccination have occurred in men 50 years old and older.

Link to this

Last July, an unpublished paper  authored  by several physicians known for spreading COVID-19 misinformation  briefly   appeared  on a preprint server hosted by the prestigious British medical journal the Lancet. 

The paper  claimed  to have reviewed autopsy reports and found — in the opinion of three of its authors — that 73.9% of the selected deaths were “directly due to or significantly contributed to by COVID-19 vaccination.” Those conclusions, however, were  often contrary  to the original scientists’ determinations. Moreover, abundant evidence contradicts the suggestion that the COVID-19 vaccines are frequently killing people.

The preprint repository quickly  removed  the manuscript because, it said, “the study’s conclusions are not supported by the study methodology,” and indicated that the preprint had violated its screening criteria. 

Social media soon flooded with posts highlighting the purported findings and alleging censorship, with many falsely stating that the paper had been published in the Lancet.

Multiple   scientists  and  fact   checkers   detailed  numerous problems with the preprint and the resulting social media posts. As Dr. Jonathan Laxton, an assistant professor of medicine at the University of Manitoba who frequently debunks misinformation online,  wrote at the time  on Twitter, “this is not a conspiracy, the paper was literally biased hot garbage and the Lancet was right to remove it.”

Despite these efforts, the same claims are back this summer after the paper was  published  in the journal Forensic Science International on June 21. Capitalizing on the paper’s now-published status,  numerous   posts   are   once   again  spreading the review’s supposed findings and realleging censorship.

“Largest autopsy series in the world. Censored by what was the most reputable peer reviewed journal,” reads  one  popular Instagram post. “74% of the 325 Suddenly Died Autopsies point the cause to the dart,” it added, using coded language to refer to the COVID-19 vaccines.

Another  post , from Dr.  Sherri Tenpenny , an osteopathic physician in Ohio known for her opposition to vaccines and her false claim that the COVID-19 vaccines magnetize people, also repeated the falsehood that the paper had been previously published in the Lancet.

“Bottom line results: 74% of sudden deaths are shown to be due to the COVID-19 vaccine,” the post went on to say. “This paper is a game changer. Sadly, it was censored for ONE YEAR. Just think of all the lives that could have been saved.”

As we’ve explained  before , publication in a peer reviewed journal does not necessarily mean a paper is accurate or trustworthy, although the process can improve manuscripts and weed out bad science. In this case, the published paper is highly similar to the previously criticized manuscript. Experts say its conclusions are unreliable and misleading.

“The vast majority of these cases do not show a causal, but coincidental, effect,” wrote Marc Veldhoen, an immunologist at the Instituto de Medicina Molecular João Lobo Antunes in Portugal, in a thread on X, addressing the paper’s central claim. “This certainly does not apply to the general population!”

When asked about the published paper, Dr. Cristina Cattaneo,  co-editor-in-chief  of Forensic Science International, told us the journal was “currently looking into the matter.”

Problematic ‘Review’

For their “ review ,” the authors searched the medical literature for published autopsy studies related to any kind of COVID-19 vaccination. After excluding duplicates and studies without deaths, autopsies, or vaccination status information, the authors were left with 44 studies comprising 325 autopsies. Three of the authors then reviewed the described cases and decided for themselves if the deaths were vaccine-related; if at least two agreed, the death was counted as being attributable to COVID-19 vaccination.

In the end, the authors thought 240, or nearly 74%, of the reviewed autopsies were vaccine-related (rounded to one decimal, 240 out of 325 is actually 73.8%, not 73.9% as reported in the paper). Among these deaths, 46.3% occurred after a Sinovac vaccine, 30.1% after a Pfizer/BioNTech vaccine, 14.6% after an AstraZeneca vaccine, 7.5% after a Moderna vaccine and 1.3% after a Johnson & Johnson vaccine.

As others have  pointed out  before, there’s reason to suspect that the authors may have been biased in their determinations. All three adjudicators, including Dr.  Peter McCullough , are well known for spreading COVID-19 misinformation. Dr. William Makis, a Canadian radiologist, has  previously claimed , without evidence, that 80 Canadian doctors died from COVID-19 vaccines. The only pathologist, Dr. Roger Hodkinson, incorrectly  claimed  in 2020 that COVID-19 was a “hoax” and “just a bad flu.”

Hodkinson and McCullough, along with five other authors, are also affiliated with and have a financial interest in The Wellness Company, a supplement and telehealth company that  sells unproven treatments , including for purported protection against vaccines.

Perhaps most tellingly, the scientists who conducted many of the autopsy studies came to opposite conclusions than the review authors. Of the 240 cases, for example, 105 come from a single  paper  in Colombia, whose authors found “[n]o relation between the cause of death and vaccination.”

Similarly, the review authors counted 24 of 28 autopsies from a  study  from Singapore as vaccine-related, even though the original authors identified “no definite causative relationship” to mRNA vaccines.

The authors of a German  study  also attributed 13 of 18 autopsy deaths to preexisting diseases, but the review authors decided 16 cases were vaccine-related.

In a  LinkedIn post  debunking the preprint, Dr.  Mathijs Binkhorst , a Dutch pediatrician, went back to each cited paper, and found that of the 325 autopsies and one heart necropsy the review authors said were vaccine-related, only 31, or 9.5%, were likely related and 28, or 8.6%, were possibly related. The rest — 267, or 81.9% — were unlikely, uncertainly, or not related to vaccination.

In other words, even among a set of studies that is more likely to identify some vaccine involvement, less than a fifth of deaths were possibly or likely vaccine-related.

Even if the authors aren’t biased, this type of study is not able to provide information on how frequently COVID-19 vaccination leads to death, and whether the risks outweigh the benefits.

“They only looked at ‘published autopsy and necropsy reports relating to COVID-19 vaccination,’” Veldhoen  said  of the published study on X. “If you look only at autopsies of those related (in time) with drugX: X-involvement is then a high proportion of all cases.”

Indeed, as Binkhorst noted, the autopsy reports come from 14 countries that collectively administered some 2.2 billion vaccine doses. If the COVID-19 vaccines truly were as dangerous as the review authors contend, this would be evident in other data sources — but it’s not.

Vaccine safety surveillance systems and other studies from across the globe have found that serious side effects can occur, but they are rare. 

The Johnson & Johnson and AstraZeneca vaccines, for example, can in very rare cases cause a dangerous and sometimes fatal blood clotting condition combined with low blood platelets. 

Rarely, the mRNA COVID-19 vaccines from Moderna and Pfizer/BioNTech have caused inflammation of the heart muscle or surrounding tissue, known as myocarditis or pericarditis. In almost all cases, however, those conditions are not deadly.

There is no evidence that COVID-19 vaccination increases the risk of death and has led to excess deaths or a large number of deaths. Instead, a wealth of data supports the notion that COVID-19 vaccines protect against severe disease and death from COVID-19. The flawed autopsy “review” doesn’t change this.

Roley, Gwen. “ Misinformation swirls around unpublished paper on Covid-19 vaccine risks .” AFP. 14 Jul 2023.

Hulscher, Nicolas et al. “ A Systematic REVIEW of Autopsy findings in deaths after covid-19 vaccination .” Forensic Science International. Available online 21 Jun 2024.

Binkhorst, Mathijs. “ McCullough’s misinformation .” LinkedIn post. Archived 4 Sep 2023.

Laxton, Jonathan (@dr_jon_l). “ McCullough et al attempted upload a preprint to the Lancet server, and it was removed because it was hot garbage.  However, I feel going through this paper for you guys will help you spot dodgy science … ” X. 6 Jul 2023.

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