Hence, the labels of the vertices \( v_{2},v_{3}, v_{4},\cdots, v_{\frac {n}{2}}~\) are n+24,n+34,n+44,,6n+4, respectively, and the labels of the vertices \( v_{\frac {n}{2}+1}, v_{\frac {n}{2}+2}, \cdots, v_{n-1},v_{n} \) are 2n,2n+10,,7n20, respectively, which are even and distinct numbers. 4, we present an edge even graceful labeling of the graphs K1+ W9, K1+ W11, K1+ W5 and K1+ W7. Finally, \( ~f^{\ast }(v_{\frac {n+1}{2}}) = [ f (e_{\frac {n+1}{2}})+ f(a_{\frac {n+}{2}})] ~\mbox{{mod}}~(4n+2) ~ =~3n+1\). {n^{2}- (\frac{n-3}{2})}\\ \end{array}\right) \end{aligned}} $$, \(~j~\longrightarrow ~~\mbox{{odd}}~~~~~ ~~\& ~~j\leq ~\frac {n-3}{2},~\), \(~~~~~j~\longrightarrow ~~\mbox{{even}}~~~~~ ~~\& ~~j\leq ~\frac {n-1}{2},\), \(~j~\longrightarrow ~~\mbox{{even}}~~~~\& ~~ ~~\frac {n+3}{2} \leq ~j~\leq n-1,\), \(~~~~~~~~~~~~~X(i,j)=~n^{2}-(\frac {n+1}{2})+j.\), \(~j~\longrightarrow ~~\mbox{{odd}}~~~~\& ~~ ~~\& ~~\frac {n+5}{2} \leq ~j~\leq ~n \), \(~~~~~~~~~~~~~X(i,j)=~n^{2}+(\frac {n+3}{2})-j.~ ~\), \(~ v_{1},v_{2},v_{3},v_{4}, \cdots, v_{\frac {n-3}{2}},v_{\frac {n-1}{2}} \), \(~\textstyle \frac {5n-1}{2}, \textstyle ~\frac {4n^{2}-5m+1}{2}, \textstyle ~\frac {9n-1}{2}, \textstyle ~\frac {4n^{2}-9m+1}{2},\cdots,\textstyle \frac {n^{2}+1}{2}, \textstyle \frac {3n^{2}-1}{2},~\), \( ~ f^{\ast }(v_{\frac {n+1}{2}})= \textstyle [~\frac {n^{3}-2n^{2}+5n-4}{2}~ ] ~\mbox{{mod}}~(2n^{2})\), \(~~~~~~~~~~~~~~~~~~~=\textstyle [~\frac {- n^{2}+5n-4}{2}~ ] ~\mbox{{mod}}~(2n^{2})=\textstyle ~\frac {3 n^{2}+5n-4}{2} \), \( ~f^{\ast }(v_{n})= [~\sum _{i=1}^{n} ~f (u_{i}~v_{n}) ~ ] ~\mbox{{mod}}~(2n^{2}) = f(u_{n} ~v_{n}) = ~\frac { 2n^{2}-n+3}{2}~ \), \( ~~~ f^{\ast }(v_{n-1})= [~\frac {-n^{3}+3n^{2}+2n-4}{2}~ ] ~\mbox{{mod}}~(2n^{2}) ~=~ n^{2}+n-2\), \( ~~~ f^{\ast }(v_{n-2})= [~\frac {n^{3}+n^{2}-2n+8}{2}~ ] ~\mbox{{mod}}~(2n^{2}) ~=~ n^{2}- n+4\), \( ~~~ f^{\ast }(v_{n-3})= [~\frac {-n^{3}-n^{2}+6n-12}{2}~ ] ~\mbox{{mod}}~(2n^{2}) ~=~ n^{2}+3n-6\), \( ~~~ f^{\ast }(v_{n-4})= [~\frac {n^{3}+n^{2}-6n+16}{4}~ ] ~\mbox{{mod}}~(2n^{2}) ~=~ n^{2}-3n+8\), \(~~ f^{\ast }(v_{n-i}) = \left \{\begin {array}{ll} ~n^{2}+in+2i ~~~ & \text {if }~~ i ~=1,3,\cdots, \frac {n-3}{2} ~; ~ \\ n^{2}-(i-1)n+2i~ ~~~~~ & \text {if}~~ ~ i =2,4,\cdots, \frac {n-4}{2}. PubMedGoogle Scholar. \end{aligned} $$, $$\begin{aligned} &\hspace{1cm} {v_{1}} \hspace{.3cm} {v_{2}} \hspace{.4cm} {v_{3}} \hspace{.4cm} {v_{4}} \hspace{.3cm} {v_{5}} \hspace{.3cm} {v_{6}} \hspace{.3cm} {v_{7}} \hspace{.3cm} {v_{8} } \hspace{.3cm} {v_{9}} \hspace{.3cm} { v_{10}} \hspace{.3cm} {v_{11}} \hspace{.3cm} {v_{12}} \hspace{.3cm} {v_{13}} \hspace{.3cm} {v_{14}} \hspace{.3cm} {v_{15}}\\ &\begin{array}{c} {u_{1}}\\ {u_{2}}\\ {u_{3}}\\ {u_{4}}\\ {u_{5}}\\ {u_{6}}\\ {u_{7}}\\ \end{array} \left(\begin{array}{ccccccccccccccc} 2 & 208 & 4 & 206 & 6 & 204 & 8 & 202 & 10 & 200& 12 & 198 & 14 & 196 & 16 \\ 180 & 30 & 182 & 28 & 184 & 26 & 186 & 24 & 188 & 22& 190 & 20 & 192 & 18 & 194 \\ 32 & 178 & 34 & 176 & 36 & 174 & 38 & 172 & 40 & 170& 42 & 168 & 44 & 166 & 46 \\ 150 & 60 & 152 & 58 & 154 & 56 & 156 & 54 & 158 & 52& 160 & 50 & 162 & 45 & 164 \\ 62 & 148 & 64 & 146 & 66 & 144 & 68 & 142 & 70 & 140 & 72 & 138 & 74 & 136 & 76 \\ 120 & 90 & 122 & 88 & 124& 86 & 126 & 84 & 128 & 82 & 130 & 80 & 132 & 78 & 134 \\ 92 & 118 & 94 & 116 & 96 & 114 & 98 & 112 & 100 & 11 0& 102 & 108 & 104 &210 & 106 \\ \end{array}\right). \), Then, by the same way, we can calculate f(x),f(y) and f(v0) and prove that they are different from all the labels of the vertices vi. The following matrix X=(aij) shows the methods of labeling, where aij represents the label of the edge uivj. when n is odd. \), \( ~~~~~~~~~~~~~~~f(y ~v_{i}~)= \left \{\begin {array}{ll} ~ ~ 6n+4~~~~~~ & \text {if} ~~ i=~0 ~; ~~\\ ~ n+1+2i ~~~~~~~~ & \text {if}~~ 1 \leq i \leq \frac {n+1}{2} ~;~~ \\ ~ 3n+1+2i~~~~~~~~~~ & \text {if} ~~ \frac {n+3}{2} \leq i \leq ~~ n ~,~~~~ ~\\ \end {array}\right. 4. We propose a graph join between two graphs G 1 1 op G 2, where the vertices are considered as to-be-joined relational tuples (V 1 1 V 2), and the resulting edges are given by combining the graph operand's edges E 1 and E 2 Discret. \( f^{\ast }(x)= \left [\sum _{i=1}^{n} f (xv_{i}) \right ]~ \mbox{{mod}}(6n)=\left [\sum _{i=1}^{n} (2i) \right ]~ \mbox{{mod}}(6n)=(n^{2}+n)\mbox{{mod}}(6n) \). For n2 (mod 6), we define the labeling function f as follows: \( ~f~(x~v_{i}) =\left \{\begin {array}{ll} ~2i ~~~~~ & \text {if}~~ 1 \leq i \leq \frac {n}{2} ~;~ ~ ~~~~~~\\ ~4n+2i~~~ ~~& \text {if }~~ \frac {n}{2} < i \leq ~~ n~,~~~~~ \\ \end {array}\right.\), \( ~~f~(v_{i}~v_{i+1}) = \left \{\begin {array}{ll} ~3n+2i ~~~ ~~~~& \text {if}~~ 1 \leq i \leq n-1 ;~~ ~\\ ~n+2 ~~~~~~~ & \text {if} ~~ i = ~~ n,~ \\ \end {array}\right.\), \(~f~(v_{0}~v_{i}) = \left \{\begin {array}{ll} ~5n ~~~ & \text {if}~~ i=1 ~;~ ~ ~~~~~\\ ~3n+4-2i~~~ & \text {if} ~~ 2 \leq i \leq ~ n~.~~~~~ \\ \end {array}\right.\), In view of the above labeling patten and since n2 (mod 6) n=6k+2 2q=6n+2=36k+14 then the induced vertex labels are, f(v0)= (2n2+5n2) mod (6n+2)=[ 2k(36k+14)+(94k+68) ] mod (36k+14), \(~~~~~~~~\equiv (14k+2) ~\mbox{{mod}}~(36 k+14~) =(\frac {7n-8}{3}). Hence, the graph K1+K1,n is edge even graceful for all n. , Illustration: In Fig. A graph consists of a set of elements together with a binary relation defined on the set. It is a commutative operation (for unlabelled graphs); [2] graph products based on the cartesian product of the vertex sets: cartesian graph product: it is a commutative and associative operation (for unlabelled graphs), [2] It should be noted that the join graph is not necessarily an edge even graceful graph. Let us use the standard notation p=|V(K1+ sfn)|=2n+1 and q =|E(K1+ sfn)|=5n. \( f^{\ast }(x)= [~ f(x~ v_{0})~+\sum _{i=1}^{n} f (x~ v_{i}~) ~ ] ~\mbox{{mod}}~(6n+4)=~ f(x ~v_{0}~)~\mbox{{mod}}~(6n+4)=~0\). ~ \), And f(vi)=[f(vivi+1)+f(vi1vi)+f(v0vi)+f(xvi)] mod(6n+2), \(~~~~~~~~~~= \left \{\begin {array}{ll} (3 n+4i+1) \mbox{{mod}} (6n+2) ~~ & \text {if}~~~~2 \leq i \leq \frac {n+1 }{2} ; \\ (n+4i-3) \mbox{{mod}} (6n+2)~ & \text {if} ~~ ~\frac {n+3 }{2} \leq i \leq n-1. https://doi.org/10.3390/sym11010038. Also, we introduced an edge even graceful labeling of the join of the graph K1 with the star graph K1,n, the wheel graph Wn and the sunflower graph sfn for all \(n \in \mathbb {N}\). else if j even, X(i,j)= 2n2[(i1)n+j+1]. 3, we present an edge even graceful labeling of the graphs K1+ W8 and K1+ W10. AIMS Mathematics, 2020, 5(6): 7214-7233. doi: 10.3934/math.2020461. -valuation introduced by Rosa [Citation 2]. Let G and H be two graphs with no vertex in common. There are two cases: when n is even. Unlike in ordinary graph theory, there are many ways to join two signed graphs. It is clear that f(x),f(y) and f(v0) are even and different from all the labels of the vertices vi. \). \), \( ~~~~~~~~ ~~~~~f(y ~v_{i}~)= \left \{\begin {array}{ll} ~ 3n+4~~~~~ & \text {if} ~~ i=~0 ~; ~~\\ ~ n+2i ~~~~~ & \text {if}~~ 1 \leq i \leq \frac {n}{2} ;~~~~ \\ ~ 3n+2+2i~~~~~~~ & \text {if} ~~ \frac {n}{2}+1 \leq i \leq ~~ n ~,~ \\ \end {array}\right.\), \(~~~~~~~~~~~~ f~(v_{0}~v_{i}) = \left \{\begin {array}{ll} ~2n+2i~~~ & \text {if}~~ 1 \leq i \leq \frac {n}{2}+1 ; \\ ~2n+2+2i~~ & \text {if} ~~ ~\frac {n}{2}+2 \leq i \leq n. \\ \end {array}\right. In this case, the equality of the labeling of the two vertices un and vn forces us to change the labels of the two edges unvn1 and unvn, that is, f(unvn1)=2n2 and f(unvn) =n2+1. \). Elsonbaty and Daoud [4] introduced a new type of labeling of a graph G with p vertices and q edges called an edge even graceful labeling if there is a bijection f from the edges of the graph to the set {2,4,, 2q } such that, when each vertex is assigned the sum of all edges incident to it mod 2k where k=max(p,q), the resulting vertex labels are distinct. https://doi.org/10.1186/s42787-019-0025-x. \( ~~~ f^{\ast }(v_{n-1})= [~\frac {-n^{3}+3n^{2}+2n-4}{2}~ ] ~\mbox{{mod}}~(2n^{2}) ~=~ n^{2}+n-2\). Let us use the standard notation \(p= | V (~\overline {K}_{2} + ~K_{1,n})| = n+3 ~\) and \(~ q~= | E (~\overline {K}_{1} + ~K_{1,n})| =3n+2~\). If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 65, 562570 (1977). If n3 (mod 6) 2q=6n=36k+18, then f(x)= [ 24k+12] mod (36k+12)= 4n. Citation: Jahfar T K, Chithra A V. Central vertex join and central edge join of two graphs[J]. - 181.119.146.47. The labels of the vertices ui, i=1,2,,n is the sum of rows in the matrix, i.e.. \(~f^{\ast }(u_{i})= [~\sum _{j=1}^{n-1} ~f (u_{i}v_{j}) ~ ] ~\mbox{{mod}}~(2n^{2}) = f(u_{i} v_{n}),~ \) so the labels of the vertices u1,u2,u3,u4,,un2,un1 are n+1, 2n2n1, 3n+1, 2n23n1,,n22n+1,n2+2n1, respectively, and f(un)=0. By a graph . \). Given a connected graph G with n vertices and given a tuple of m = { m 1, m 2,., m n } non-negative integers, we form a new graph G ( m) by considering the complete graph K m i for each vertex i and 'join' two of such complete graphs if the corresponding vertices are adjacent. PDF downloads(154) else if \(~j~\longrightarrow ~~\mbox{{even}}~~~~\& ~~ ~~\frac {n+3}{2} \leq ~j~\leq n-1,\)\(~~~~~~~~~~~~~X(i,j)=~n^{2}-(\frac {n+1}{2})+j.\) else if \(~j~\longrightarrow ~~\mbox{{odd}}~~~~\& ~~ ~~\& ~~\frac {n+5}{2} \leq ~j~\leq ~n \),\(~~~~~~~~~~~~~X(i,j)=~n^{2}+(\frac {n+3}{2})-j.~ ~\). IEEE. Given two signed graphs, the chromatic number of the all-positive and all-negative join is usually less than the sum of their chromatic numbers, by an amount that depends on the new concept of deficiency of a signed-graph coloration. \\ \end {array}\right. \), $$\begin{aligned} f^{\ast}(x)&= \left[ f(x~ v_{0})~+\sum_{i=1}^{n} f (x~ v_{i}) \right] {\rm{mod}}~(6n+4)= f(x ~v_{0}~)~{\rm{mod}}~(6n+4)=~0,\\ ~~~f^{\ast}(y)&= \left[ f(y ~v_{0})~+\sum_{i=1}^{n} f (y ~v_{i}~) ~ \right] ~{\rm{mod}}~(6n+4)=~ f(y ~ v_{0}~)~{\rm{mod}}~(6n+4)=2,\\ f^{\ast}(v_{0}) &= \left[ \sum_{i=1}^{n} f (v_{0}v_{i}) + f(xv_{0})+f(yv_{0})\right] {\rm{mod}}(6n+4) = ~4, \end{aligned} $$, \( v_{1},~v_{2},~ \cdots, ~ v_{\frac {n-3}{2}}, ~ v_{\frac {n-1}{2}}~~\), \(~v_{\frac {n+1}{2}},~ v_{\frac {n+3}{2}},~~ \cdots, v_{n-2},~v_{n-1},~v_{n} \), \(~ E(\overline {K}_{2} + ~W_{n})= \{ x~v_{0},~ x~v_{i},~y~v_{0},y~v_{i},v_{0}~v_{i}~,v_{i}~v_{i+1},~~i=1,2, \cdots,~n~\}\), \(p = | V (~\overline {K}_{2} + ~w_{n})| = n+3 \), \( q~= | E (~\overline {K}_{2} + ~W_{n})| =4n+2~\), \(f:E(\overline {K}_{2} + ~W_{n})\longrightarrow \{2,4,\cdots,8n+4\}\), \( f(x ~v_{i}~)= \left \{\begin {array}{ll} ~2~~~ & \text {if }~~ i=~0 ~; \\ ~n+2+2i ~~~ & \text {if}~~ 1 \leq i \leq \frac {n}{2} ; \\ ~5n+2i ~~~ & \text {if}~~ \frac {n}{2} < i \leq ~~ n, \\ \end {array}\right. volume28, Articlenumber:21 (2020) \end{aligned} $$, $$\begin{aligned} f^{\ast}(x)&= [~ f(x~ v_{0})~+\sum_{i=1}^{n} f (x v_{i}~) ~ ] ~{\rm{mod}}~(4n+2)=~ f(x ~v_{0}~)~{\rm{mod}}~(4n+2)=~2n,\\ f^{\ast}(v_{0})&= [~ f(x ~v_{0})~+\sum_{i=1}^{n} f (v_{0}~v_{i}~) ~ ] ~{\rm{mod}}~(4n+2)\\&=~ [f(x ~ v_{0}~)+ f(a_{1})]~{\rm{mod}}~(4n+2)=~0,\\ ~f^{\ast}(v_{i})&= [ f (x v_{i})+ f(v_{0}~v_{i})] ~{\rm{mod}}~(4n+2) \end{aligned} $$, \( v_{1},v_{2},~ \cdots, v_{\frac {n}{2}-1},v_{\frac {n}{2}}~\), \(~v_{\frac {n}{2}+1},~ v_{\frac {n}{2}+2},~v_{\frac {n}{2}+3},~ \cdots,~ v_{n-1},~v_{n} \), \(~ f^{\ast }(v_{\frac {n}{2}+1}) = [ f (x~v_{\frac {n}{2}+1})+ f(v_{0}~v_{\frac {n}{2}+1})] ~\mbox{{mod}}~(4n+2) =~3n+2 \), \(f^{\ast }(v_{0})= [ f(x ~v_{0})~+\sum _{i=1}^{n} f (v_{0}v_{i}~) ] ~\mbox{{mod}}~(4n+2)=~n+1,~~~ f^{\ast }(x)= ~ 0~\), \( v_{1},v_{2},~ \cdots, v_{\frac {n-1}{2}},v_{\frac {n+1}{2}} ~ \), \(v_{\frac {n+3}{2}}, v_{\frac {n+5}{2}},~ \cdots,~ v_{n-1},v_{n} \), \( ~f^{\ast }(v_{\frac {n+1}{2}}) = [ f (e_{\frac {n+1}{2}})+ f(a_{\frac {n+}{2}})] ~\mbox{{mod}}~(4n+2) ~ =~3n+1\), \(\protect \phantom {\dot {i}\! MathSciNet For |V (G)| = m and |V (H)| = n, theedgesetofG+H is the union of disjoint edge sets of the graphs G, H,and the complete bipartite graph K m,n . Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Google Scholar. \). For n1 (mod 6), we defined the labeling function f as follows: f(v1vn)=4n, f(vivi+1)=2n+2ifori=1,2,n1, \( f~(x~v_{i}) = \left \{\begin {array}{ll} 2i~~~ & \text {if} ~~ 1 \leq i \leq ~\frac {n+1}{2}; \\ 4n-2+2i ~~~ & \text {if}~~ \frac {n+3}{2} \leq i \leq n, \\ \end {array}\right. Your US state privacy rights, If n1 (mod 6) 2q=6n=36k+6, then f(x)[ 12k+2] mod (36k+6)= 2n. All common graph operations and more are built into Mathematica 8. In this graph, \(p = | V (~\overline {K}_{2} + ~w_{n})| = n+3 \) and \( q~= | E (~\overline {K}_{2} + ~W_{n})| =4n+2~\). and \(~~~ f^{\ast }(v_{\frac {n}{2}+1}) = \left [ f \left (v_{0}v_{\frac {n}{2}+1}\right)+ f\left (x~v_{\frac {n}{2}+1}\right)+f\left (y~v_{\frac {n}{2}+1}\right)\right ] ~\mbox{{mod}}~(6n+4) ~ =~2\). The join product of two graphs G and H, denoted by G + H, is obtained from vertex-disjoint copies of G and H by adding all edges between V (G)andV (H). View full-text Article To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Method 1: We define the labeling function f as follows: \( ~~~~~~~~~~~~~~~f~(v_{0}~v_{i}) =\left \{\begin {array}{ll} ~2n+2+2i~~~~~ & \text {if}~~ 1 \leq i \leq \frac {n-1}{2} ; \\ ~5n+1-2i~~~ & \text {if} ~~ ~\frac {n+1}{2} \leq i \leq ~n-1 ; \\ ~6n+2~~~ & \text {if} ~~ ~ i =~n~,~~ \\ \end {array}\right. The join of G and H, denoted by G+H, defined to be the graph with vertex set and edge set given as follows: V(G+H)=V(G) V(H),E(G+H)=E(G)E(H){x1x2:x1V(G), x2V(H)}. Furthermore, Elsonbaty and Daoud [6] investigate edge even graceful labeling of cylinder grid graphs also, Daoud [7] studied the edge even graceful labeling of Polar grid graphs after that, Zeen El Deen and Omar N. [8] extended the edge even graceful labeling into r- edge even graceful labeling. When n=5, the graph E(K1+ W5) is an edge even graceful graph but it does not follow this rule, see Fig. \vdots & \!\!\vdots& \!\!\ldots & \!\!\ldots & \!\!\vdots &\!\!\ddots & \ldots & \ldots \\ \!\! If n0 (mod 6) n=6k 2q=6n+2=36k+2, then. \), \(~~ f^{\ast }(v_{0})= [~\sum _{i=1}^{n} f (v_{0}~v_{i}) +f(v_{0}~x)~ ] ~\mbox{{mod}}~(6n+2)=~0~\), \(~~~~~~~~~~~=[\sum _{i=1}^{n} (3n+2i)~+(6n+2) ] ~\mbox{{mod}}~(6n+2)=(4n^{2}+n)~\mbox{{mod}}(6n+2)\), $$\begin{array}{*{20}l} f^{\ast}(x) &= [~4~(6k)^{2}+ (6k) ~]~ {\rm{mod}}~(36 k+2~)~\\&=[ ~4k (36k+2) - (2k)~ ]~ {\rm{mod}}~(36 k+2~)~ \\ &\equiv ~(-2k) ~{\rm{mod}}~(36 k+2~) \equiv (34k+2)~{\rm{mod}}~(36 k+2~) \\&\quad=~\left(\frac{17n+6}{3} \right). else if j ==n, X(i,j)= 2n2[(i1)n+1]. \( ~~~ f^{\ast }(v_{n-1})= \textstyle [~\frac {-n^{3}+n^{2}+n-1 }{2}~ ] ~\mbox{{mod}}~(2n^{2}) =\textstyle [~\frac {- 2n^{2}+n-1}{2}~ ] ~\mbox{{mod}}~(2n^{2}) ~=~\frac {n^{2}+n-1}{2}\). The graph that admits edge even graceful labeling is called an edge even graceful graph. Here, p=|V(K1+K1,n)|=n+2 and q =|E(K1+K1,n)|=2n+1. \\ \end {array}\right. The SG vertex join of G1 and G2 is denoted by G1 G2 and is the graph obtained from S (G1)G1 and G2 by joining every vertex of V (G1) to every vertex of V (G2). under the terms of the Creative Commons Attribution License Oct 31, 2021 5: Graph Theory 5.2: Euler Circuits and Walks David Guichard Whitman College See Section 4.5 to review some basic terminology about graphs. We introduce a conjecture and open problems for researchers for furtherresearch.Keywords:edge irregularity strength, blocks, chain graphs, join of graphsMathematics Subject Classication : 05C78, 05C12DOI: 10.5614/ejgta.2018.6.1.15 Finally,\( ~~~~~f^{\ast }(v_{i}) = \left \{\begin {array}{ll} (5n+3+2i) ~\mbox{{mod}}~(8n+4)~~~ & \text {if}~~ 1 \leq i \leq \frac {n-1 }{2}; \\ (n-7+2i) ~\mbox{{mod}}~(8n+4) ~~~ & \text {if} ~~~~ \frac {n+3 }{2} \leq i \leq n-1.\\ \end {array}\right.\). \), \( f^{\ast }(x)= \left [~ f(x~ v_{0})~+\sum _{i=1}^{n} f (x~ v_{i}~) ~ \right ] ~\mbox{{mod}}~(8n+4)=~ f(x ~v_{0}~)~\mbox{{mod}}~(8n+4)=~2\), \( f^{\ast }(y)= \left [ f(y v_{0})+\sum _{i=1}^{n} f (y v_{i}) \right ] ~\mbox{{mod}}(8n+4)=~f(y v_{0})~\mbox{{mod}}(8n+4)=8n+2\), \( f^{\ast }(v_{0}) = \left [ \sum _{i=1}^{n} f (v_{0}~v_{i}) + f(x~v_{0})+f(y~v_{0})\right ] ~\mbox{{mod}}~(8n+4) ~ =~0\), \(~~~~~f^{\ast }(v_{i}) = \left \{\begin {array}{ll} (n+4+10i) ~\mbox{{mod}}~(8n+4)~~~ & \text {if}~~ ~2 \leq i \leq \frac {n }{2}; \\ (5n-6+10i) ~\mbox{{mod}}~(8n+4) ~~~ & \text {if} ~~~~~ \frac {n }{2}+1 \leq i < n. \\ \end {array}\right. Springer Nature. Graphs $ P_2 \dot{\vee} P_3 $, $ P_3 \dot{\vee} P_2 $, $ P_2\veebar P_3 $ and $ P_3 \veebar P_2 $. https://doi.org/10.1007/s00373-021-02387-6, access via Similarly, \( ~~f^{\ast }(y)= [~\sum _{i=1}^{n} f (yv_{i}) ~ ] ~\mbox{{mod}}~(~6n)=\left [~\sum _{i=1}^{n} (6n-2i) \right ]~ \mbox{{mod}}~(6n) \). Daoud, S. N.: Edge even graceful labeling of polar grid graphs. Hypergraphs are able to capture interactions between two or more nodes and for this reason they are raising significant attention in the context of opinion dynamics, epidemic spreading, synchronization and game theory. volume37,pages 27232735 (2021)Cite this article. Finally, we proved that the join of the graph \(\overline {K}_{2}~\) with the star graph K 1,n, the wheel graph W n and the cyclic graph C n are edge even graceful graphs. The join graph K1+ Wn has an edge even graceful labeling for all n. Let {v0,v1,v2, , vn} be the vertices of Wn with central vertex v0 and {x} be the vertex of K1 so the edges of K1+ Wn will be {xv0, xvi, v0vi,vivi+1,i=1,2,, n }. For graph theoretic terminology we refer to West [Citation 1]. Hence, the labels of the vertices \( v_{1},v_{2},v_{3}, \cdots,~v_{\frac {n-3}{2}},~ v_{\frac {n-1}{2}}~~\) are 6,3n+11,3n+15,,5n3,5n+1, respectively, and the labels of the vertices \(~ v_{\frac {n+1}{2}},~ v_{\frac {n+3}{2}},~ ~ \cdots, v_{n-1},v_{n} ~~\) are 3n+1, 3n+5, , 5n5, 6n, respectively. your institution. (Submitted), Zaslavsky, Thomas: Signed graph coloring. Cartesian product of graphs - the recognition problem #. We define the labeling function \(~f:E(\overline {K}_{2} + ~K_{1,n})\longrightarrow \{2,4,\cdots,6n+4\}\) as follows: \(~~~~~~~~~~~~~f(x ~v_{i}~)= \left \{\begin {array}{ll} 6n+4~~~ & \text {if} ~~ i=~0 ~; \\ 2i ~~~~ & \text {if}~~ 1 \leq i \leq \frac {n}{2} ; \\ 4n+2+2i ~~~~ & \text {if}~~ \frac {n}{2}+1 \leq i \leq ~~ n, \\ \end {array}\right. \). Google Scholar. We define the labeling \(f:E(\overline {K}_{2} + ~C_{n})\longrightarrow \{2,4,\cdots,6n\}\) as follows: f(v1vn)=6n, f(vivi+1)=2n+2ifori=1,2,n1. MATH \), \(~~~~~~~~ f~(u_{i}~v_{j}) = \left \{\begin {array}{ll} ~(i-1)~n~+j+2~~~~ & \text {if}~~ j = ~1,3,\cdots, ~n-2 ; \\[-3pt] \par ~2n^{2}- [(i-1)n+j+1]~~~ & \text {if }~~ ~j=2,4,\cdots, ~n-1 ; \\[-3pt] \par ~2n^{2}- [(i-1)n+1]~~~ & \text {if }~~ ~j=n. So to allow loops the definitions must be expanded. The eccentric graph of eccentric join of graphs is also examined. and f(vn)=[f(v0vn)+f(xvn)+f(yvn)] mod (6n+4) = 6n2. \\ \end {array}\right. Thus, the graph \(~ \overline {K}_{2} + ~W_{n} \) is an edge even graceful graph for all n. . f(vn)=[f(vnv1)+f(vn1vn)+f(xvn)+f(yvn)] mod (6n)= 4n2. For example, the wheel graphW3=K1 +C3 is not an edge even graceful graph. Definition The Cartesian product of two graphs G and H, denoted G H, is a graph defined on the pairs ( g, h) V ( G) V ( H). \( \therefore ~~~ f^{\ast }(y)=(5 n^{2}-n)~\mbox{{mod}}(6n) =\left \{\begin {array}{ll} 4n ~~~ & \text {if}~~ n \equiv 1~(\mbox{{mod}}~ 6) ;\\ 2n~~~ & \text {if} ~~ n \equiv 3~(\mbox{{mod}}~ 6)~. Journal of the Egyptian Mathematical Society Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Hence, the labels of the vertices \( v_{1},v_{2},v_{3}, \cdots, v_{\frac {n}{2}}~\) will be n+4,n+8,n+12,,3n, respectively, and the labels of the vertices \( v_{\frac {n}{2}+1}, v_{\frac {n}{2}+2}, \cdots, v_{n-1},v_{n} ~\) will be n+2,n+6,,3n6,3n2, respectively, which are all even and distinct numbers. Let 'G' = (V, E) be a graph. 9, we present an edge even graceful labeling of \(\overline {K}_{2} + ~C_{7}, \overline {K}_{2} + ~C_{9}, \overline {K}_{2} + ~C_{11}\) and \(\overline {K}_{2} + ~C_{12}\), An edge even graceful labeling of the double cones \( \overline {K}_{2} + ~C_{7} \), \( \overline {K}_{2} + ~C_{9} \), \( \overline {K}_{2} + ~C_{11} \), and \( \overline {K}_{2} + ~C_{12}\). \), \( ~~ ~f(y ~v_{i}~)= \left \{\begin {array}{ll} ~8n~~~ & \text {if} ~~ i=~0 ~; \\ ~2n+2+2i ~~~ & \text {if}~~ 1 \leq i \leq \frac {n}{2} ;~ \\ ~4n+2i ~~~ & \text {if}~~ \frac {n}{2} < i \leq ~~ n.~ \\ \end {array}\right. { xvi, xui, vivi+1, viui, vi+1ui, i=1,2,,n }. The vertices of Kn,n were divided into two disjoint sets {v1,v2,,vn} {u1,u2,,un} such that every pair of graph vertices in the two sets are adjacent. MATH Obviously the vertex labels are all even and distinct. Gross, J., Yellen, J.: Graph theory and its applications. We define the labeling function f:E(K1+K1,n){2,4,,4n+2} as follows: \(~ f~(x~v_{i}) = \left \{\begin {array}{ll} 2i~~~ & \text {if }~~ i=1,2,\cdots ~, \frac {n}{2} ~; \\ 2n+2i ~~~ & \text {if}~~ i =~\frac {n}{2}+1,\cdots, n ~.~ ~ \\ \end {array}\right. \(~~~~~~~\equiv \left [~24(\frac {n-5}{6})+ 20 ~\right ]~ \mbox{{mod}}~(6 n~)= ~4n\). Correspondence to Obviously, the vertex labels are all even and distinct. The labels of the edges \( u_{n}v_{1},~u_{n}v_{2},~u_{n}v_{3},~u_{n}v_{4},~\cdots,~ u_{n}v_{\frac {n-3}{2}},~u_{n}v_{\frac {n-1}{2}} \) are given by \( ~ (n-1)n+2, ~ 2n^{2}-[(n-1)n+2],~(n-1)n+4,~2n^{2}-[(n-1)n+4], ~\cdots, ~n(n-1)+(\frac {n-1}{2}),~2n^{2}-[n(n-1)+(\frac {n-1}{2}) ] \) and the edge \( ~ u_{n}v_{\frac {n+1}{2}}~\) label by 2n2. More precisely, we prove that the join of two graphs admits a nowhere-zero 3-flow except for a few classes of graphs: a single vertex joined with a graph containing an isolated vertex or. 1. \), \(~f~(v_{0}~v_{i}) = \left \{\begin {array}{ll} 2n+2i ~~~ & \text {if}~~ i=1,2,\cdots ~, \frac {n}{2} ~;~ ~ \\ 4n+2~~~ & \text {if} ~~ i =~\frac {n}{2}+1~; \\ 2i-2~~~ & \text {if} ~~ i =~\frac {n}{2}+2,\cdots, ~ n ~. J., Yellen, J.: graph theory, there are many ways to join two graphs! [ citation 1 ] Submitted ), Zaslavsky, Thomas: signed graph coloring 24k+12 ] mod ( )! Central vertex join and Central edge join of two graphs [ j ] for theoretic... 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