Introduction
Preliminaries
Acronyms | Description |
---|---|
FS | Fuzzy set |
CFS | Complex fuzzy set |
CFON | Complex fuzzy out-neighborhood |
CFIN | Complex fuzzy in-neighborhood |
CFCG | Complex fuzzy competition graph |
\(CFO^{p}N\) | Complex fuzzy open neighborhood |
\(CFC^{l}N\) | Complex fuzzy closed neighborhood |
\(CFO^{p}NG\) | Complex fuzzy open neighborhood graph |
\(CFC^{l}NG\) | Complex fuzzy closed neighborhood graph |
CFECG | Complex fuzzy economic competition graph |
Complex fuzzy set applied to competition graphs
g |
\(\aleph ^{p}(g)\)
|
\(\aleph ^{n}(g)\)
|
---|---|---|
i |
\(\varnothing \)
|
\(\{(j,0.3e^{i0.20\pi }),(n,0.35e^{i0.10\pi })\}\)
|
j |
\(\{(l,0.50e^{i0.40\pi }),(i,0.30e^{i0.20\pi })\}\)
|
\(\varnothing \)
|
l |
\(\varnothing \)
|
\(\{(n,0.40e^{i0.90\pi }),(j,0.5e^{i0.40\pi })\}\)
|
n |
\(\{(l,0.40e^{i0.90\pi }),(i,0.35e^{i0.10\pi })\}\)
|
\(\varnothing \)
|
g |
\(\aleph ^{p}(g)\)
|
---|---|
l |
\(\left\{ \left( k,0.3e^{i0.2\pi }\right) \right\} \)
|
i |
\(\left\{ \left( l,0.2e^{i0.3\pi }\right) \right\} \)
|
m |
\(\left\{ \left( i,0.35e^{i1.1\pi }\right) \right\} \)
|
k |
\(\left\{ \left( m,0.2e^{i0.7\pi }\right) \right\} \)
|
n |
\(\left\{ \left( l,0.25e^{i0.2\pi }\right) ,\left( k,0.2e^{i0.7\pi }\right) \right\} \)
|
g | f |
\(\aleph ^{p}(g)\cap \aleph ^{p}(f)\)
|
\({\hbar }\left( \aleph ^{p}(g)\cap \aleph ^{p}(f)\right) \)
|
---|---|---|---|
l | i |
\(\varnothing \)
|
\(\varnothing \)
|
l | m |
\(\varnothing \)
|
\(\varnothing \)
|
l | k |
\(\varnothing \)
|
\(\varnothing \)
|
l | n |
\(\left\{ \left( k,0.2e^{i0.2\pi }\right) \right\} \)
|
\(\left\{ \left( 0.2e^{i0.2\pi }\right) \right\} \)
|
i | m |
\(\varnothing \)
|
\(\varnothing \)
|
i | k |
\(\varnothing \)
|
\(\varnothing \)
|
i | n |
\(\left\{ \left( l,0.2e^{i0.2\pi }\right) \right\} \)
|
\(\left\{ \left( 0.2e^{i0.2\pi }\right) \right\} \)
|
m | k |
\(\varnothing \)
|
\(\varnothing \)
|
m | n |
\(\varnothing \)
|
\(\varnothing \)
|
k | n |
\(\varnothing \)
|
\(\varnothing \)
|
g | \(\aleph ^{p}(g)\) |
---|---|
j | \(\left\{ \left( n,0.4e^{i1.0\pi }\right) ,\left( m,0.36e^{i0.2\pi }\right) ,\left( x,0.3e^{i0.2\pi }\right) ,\left( z,0.35e^{i0.9\pi }\right) \right\} \) |
l | \(\left\{ \left( n,0.35e^{i0.2\pi }\right) ,\left( m,0.45e^{i0.3\pi }\right) ,\left( x,0.6e^{i0.3\pi }\right) ,\left( z,0.5e^{i0.2\pi }\right) \right\} \) |
x | \(\varnothing \) |
z | \(\varnothing \) |
m | \(\varnothing \) |
n | \(\varnothing \) |
g | \(\aleph ^{p}(g)\) |
---|---|
m | \(\{\left( r,0.7e^{i0.8\pi }\right) ,(s,0.7e^{i1.3\pi })\}\) |
h | \(\{(r,0.6e^{i0.3\pi }),(s,0.8e^{i0.7\pi }),(z,0.8e^{i0.3\pi })\}\) |
j | \(\{(z,0.8e^{i0.3\pi })\}\) |
r | \(\varnothing \) |
s | \(\varnothing \) |
z | \(\varnothing \) |
Complex fuzzy set applied to neighborhood graphs
g |
\(\aleph (g)\)
|
\(\aleph [g]\)
|
---|---|---|
p |
\(\{(q,0.3e^{i1.7\pi })(t,0.3e^{i1.8\pi }),(s,0.3e^{i1.7\pi })\}\)
|
\(\{(q,0.3e^{i1.7\pi })((t,0.3e^{i1.8\pi }),(s,0.3e^{i1.7\pi }))\}\)
|
\(\cup \{(p,0.3e^{i1.9\pi })\}\)
| ||
q |
\(\{(p,0.3e^{i1.7\pi }),(r,0.4e^{i1.5\pi })\}\)
|
\(\{(p,0.3e^{i1.7\pi }),(r,0.4e^{i1.5\pi })\}\cup \{(q,0.6e^{i1.8\pi })\}\)
|
r |
\(\{(q,0.4e^{i1.5\pi }),(s,0.4e^{i1.4\pi })\}\)
|
\(\{(q,0.4e^{i1.5\pi }),(s,0.4e^{i1.4\pi })\}\cup \{(r,0.4e^{i1.5\pi })\}\)
|
s |
\(\{(r,0.4e^{i1.4\pi }),(t,0.4e^{i1.7\pi }),(p,0.3e^{i1.7\pi })\}\)
|
\(\{(r,0.4e^{i1.4\pi }),(t,0.4e^{i1.7\pi }),(p,0.3e^{i1.7\pi })\}\)
|
\(\cup \{(s,0.5e^{i1.7\pi })\}\)
| ||
t |
\(\{(p,0.3e^{i1.8\pi }),(s,0.4e^{i1.7\pi })\}\)
|
\(\{(p,0.3e^{i1.8\pi }),(s,0.4e^{i1.7\pi })\}\cup \{(t,0.4e^{i1.9\pi })\}\)
|
g | f |
\(\aleph (g)\cap \aleph (f)\)
|
\(\aleph [g]\cap \aleph [f]\)
| ||
---|---|---|---|---|---|
p | q |
\(\varnothing \)
|
\(\{(p,0.3e^{i1.7\pi }),(q,0.3e^{i1.7\pi })\}\)
| ||
p | r |
\(\{(q,0.3e^{i1.5\pi }),(s,0.3e^{i1.4\pi })\}\)
|
\(\{(q,0.3e^{i1.5\pi }),(s,0.3e^{i1.4\pi })\}\)
| ||
p | s |
\(\{(t,0.3e^{i1.7\pi })\}\)
|
\(\{(p,0.3e^{i1.7\pi }),(s,0.3e^{i1.7\pi }),(t,0.3e^{i1.7\pi })\}\)
| ||
p | t |
\(\{(s,0.3e^{i1.7\pi })\}\)
|
\(\{(p,0.3e^{i1.8\pi }),(s,0.3e^{i1.7\pi }),(t,0.3e^{i1.8\pi })\}\)
| ||
q | r |
\(\varnothing \)
|
\(\{(q,0.4e^{i1.5\pi }),(r,0.4e^{i1.5\pi })\}\)
| ||
q | s |
\(\{(r,0.4e^{i1.4\pi }),(p,0.3e^{i1.7\pi })\}\)
|
\(\{(p,0.3e^{i1.7\pi }),(r,0.4e^{i1.4\pi })\}\)
| ||
q | t |
\(\{(p,0.3e^{i1.7\pi })\}\)
|
\(\{(p,0.3e^{i1.7\pi })\}\)
| ||
r | s |
\(\varnothing \)
|
\(\{(s,0.4e^{i1.4\pi }),(r,0.4e^{i1.4\pi })\}\)
| ||
r | t |
\(\{(s,0.4e^{i1.4\pi })\}\)
|
\(\{(s,0.4e^{i1.4\pi })\}\)
| ||
s | t |
\(\{(p,0.3e^{i1.7\pi })\}\)
|
\(\{(p,0.3e^{i1.7\pi }),(s,0.4e^{i1.7\pi }),(t,0.4e^{i1.7\pi })\}\)
|
g | f |
\({\hbar }\left( \aleph (g)\cap \aleph (f)\right) \)
|
\({\hbar }\left( \aleph [g]\cap \aleph [f]\right) \)
|
\(|\aleph (g)\cap \aleph (f)|\)
|
\(|\aleph [g]\cap \aleph [f]|\)
|
---|---|---|---|---|---|
p | q |
\(\varnothing \)
|
\(\{(0.3e^{i1.7\pi })\}\)
|
\(\varnothing \)
|
\((0.6e^{i3.4\pi })\)
|
p | r |
\(\{(0.3e^{1.5\pi })\}\)
|
\(\{(0.3e^{i1.5\pi })\}\)
|
\((0.6e^{i2.9\pi })\)
|
\((0.6e^{i2.9\pi })\)
|
p | s |
\(\{(0.3e^{i1.7\pi })\}\)
|
\(\{(0.3e^{i1.7\pi })\}\)
|
\((0.3e^{i1.7\pi })\)
|
\((0.9e^{i5.1\pi })\)
|
p | t |
\(\{(0.3e^{i1.7\pi })\}\)
|
\(\{(0.3e^{i1.8\pi })\}\)
|
\((0.3e^{i1.7\pi })\)
|
\((0.9e^{i5.3\pi })\)
|
q | r |
\(\varnothing \)
|
\(\{(0.4e^{i1.5\pi })\}\)
|
\(\varnothing \)
|
\((0.8e^{i3.0\pi })\)
|
q | s |
\(\{(0.4e^{i1.7\pi })\}\)
|
\(\{(0.4e^{i1.7\pi })\}\)
|
\((0.7e^{i3.1\pi })\)
|
\((0.7e^{i3.1\pi })\)
|
q | t |
\(\{(0.3e^{i1.7\pi })\}\)
|
\(\{(0.3e^{i1.7\pi })\}\)
|
\((0.3e^{i1.7\pi })\)
|
\((0.3e^{i1.7\pi })\)
|
r | s |
\(\varnothing \)
|
\(\{(0.4e^{i1.4\pi })\}\)
|
\(\varnothing \)
|
\((0.8e^{i2.8\pi })\)
|
r | t |
\(\{(0.4e^{i1.4\pi })\}\)
|
\(\{(0.4e^{i1.4\pi })\}\)
|
\((0.4e^{i1.4\pi })\)
|
\((0.4e^{i1.4\pi })\)
|
s | t |
\(\{(0.3e^{i1.7\pi })\}\)
|
\(\{(0.4e^{i1.7\pi })\}\)
|
\((0.3e^{i1.7\pi })\)
|
\((1.1e^{i5.1\pi })\)
|
Complex fuzzy set applied to m-step competition graphs
g | \(\aleph ^{p}_{2}(g)\) |
---|---|
a | \(\{\left( x,0.6e^{i0.5\pi }\right) ,\left( y,0.4e^{i0.3\pi }\right) \}\) |
b | \(\{\left( x,0.6e^{i1.0\pi }\right) ,\left( y,0.3e^{i0.3\pi }\right) \}\) |
c | \(\varnothing \) |
x | \(\varnothing \) |
y | \(\varnothing \) |
z | \(\varnothing \) |
g | \(\aleph _{2}(g)\) |
---|---|
a | \(\{(c,0.7e^{i0.7\pi }),(d,0.7e^{i0.6\pi })\}\) |
b | \(\{(f,0.7e^{i0.7\pi }),(d,0.8e^{i0.5\pi })\}\) |
c | \(\{(f,0.7e^{i0.5\pi }),(e,0.6e^{i0.5\pi }),(a,0.7e^{i0.7\pi })\}\) |
d | \(\{(a,0.7e^{i0.6\pi }),(b,0.8e^{i0.5\pi })\}\) |
e | \(\{(c,0.6e^{i0.5\pi }),(f,0.6e^{i0.6\pi })\}\) |
f | \(\{(e,0.6e^{i0.6\pi }),(c,0.7e^{i0.5\pi }),(b,0.7e^{i0.7\pi })\}\) |
g | f | \(\aleph (g)\cap \aleph (f)\) | \({\hbar }(\aleph (g)\cap \aleph (f))\) |
---|---|---|---|
a | b | \(\{(d,0.7e^{i0.5\pi })\}\) | \(\{(0.7e^{i0.5\pi })\}\) |
a | c | \(\varnothing \) | \(\varnothing \) |
a | d | \(\varnothing \) | \(\varnothing \) |
a | e | \(\{(c,0.6e^{i0.5\pi })\}\) | \(\{(0.6e^{i0.5\pi })\}\) |
a | f | \(\{(c,0.7e^{i0.5\pi })\}\) | \(\{(0.7e^{i0.5\pi })\}\) |
b | c | \(\{(f,0.7e^{i0.5\pi })\}\) | \(\{(0.7e^{i0.5\pi })\}\) |
b | d | \(\varnothing \) | \(\varnothing \) |
b | e | \(\{(f,0.6e^{i0.6\pi })\}\) | \(\{(0.6e^{i0.6\pi })\}\) |
b | f | \(\varnothing \) | \(\varnothing \) |
c | d | \(\{(a,0.7e^{i0.6\pi })\}\) | \(\{(0.7e^{i0.6\pi })\}\) |
c | e | \(\{(f,0.6e^{i0.5\pi })\}\) | \(\{(0.6e^{i0.5\pi })\}\) |
c | f | \(\{(e,0.6e^{i0.5\pi })\}\) | \(\{(0.7e^{i0.5\pi })\}\) |
d | e | \(\varnothing \) | \(\varnothing \) |
d | f | \(\{(b,0.7e^{i0.5\pi })\}\) | \(\{(0.7e^{i0.5\pi })\}\) |
e | f | \(\{(c,0.6e^{i0.5\pi })\}\) | \(\{(0.6e^{i0.5\pi })\}\) |
Complex fuzzy economic competition graphs
x | \(\aleph ^{n}(x)\) | \(\aleph ^{n}_{2}(x)\) |
---|---|---|
g | \(\{(f,0.6e^{i1.2\pi })\}\) | \(\{a,0.6e^{i1.2\pi }\}\) |
f | \(\{(a,0.7e^{i1.2\pi })\}\) | \(\varnothing \) |
a | \(\varnothing \) | \(\varnothing \) |
b | \(\{(a,0.7e^{i1.1\pi })\}\) | \(\varnothing \) |
c | \(\{(d,0.5e^{i0.8\pi }),(b,0.6e^{i1.0\pi })\}\) | \(\{(a,0.6e^{i1.0\pi })\} ~or ~\{(a,0.5e^{i0.8\pi })\}\) |
e | \(\{(f,0.7e^{i0.6\pi }),(d,0.6e^{i0.7\pi })\}\) | \(\{(a,0.7e^{i0.6\pi })\}~or~\{(a,0.6e^{i0.7\pi })\}\) |
d | \(\{(a,0.6e^{i0.8\pi })\}\) | \(\varnothing \) |
a | b | \(\aleph ^{n}(a)\cap \aleph ^{n}(b)\) | \(\aleph ^{n}_{2}(a)\cap \aleph _{2}^{n}(b)\) | \({\hbar }(\aleph ^{n}(a)\cap \aleph ^{n}(b))\) | \({\hbar }(\aleph ^{n}_{2}(a)\cap \aleph ^{n}_{2}(b))\) |
---|---|---|---|---|---|
g | f | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
g | a | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
g | b | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
g | c | \(\varnothing \) | \(\{(a,0.5e^{i0.8\pi })\}\) | \(\varnothing \) | \(\{(0.5e^{i0.8\pi })\}\) |
g | e | \(\{(f,0.6e^{i0.6\pi })\}\) | \(\{(a,0.6e^{i0.6\pi })\}\) | \(\{(0.6e^{i0.6\pi })\}\) | \(\{(0.6e^{i0.6\pi })\}\) |
g | d | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
f | a | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
f | b | \(\{(a,0.7e^{i1.1\pi })\}\) | \(\varnothing \) | \(\{(0.7e^{i1.1\pi })\}\) | \(\varnothing \) |
f | c | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
f | e | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
f | d | \(\{(a,0.6e^{i0.8\pi })\}\) | \(\varnothing \) | \(\{(0.6e^{i0.8\pi })\}\) | \(\varnothing \) |
a | b | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
a | c | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
a | e | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
a | d | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
b | c | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
b | e | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
b | d | \(\{(a,0.6e^{i0.8\pi })\}\) | \(\varnothing \) | \(\{(0.6e^{i0.8\pi })\}\) | \(\varnothing \) |
c | e | \(\{(a,0.5e^{i0.7\pi })\}\) | \(\{(a,0.5e^{i0.7\pi })\}\) | \(\{(0.5e^{i0.7\pi })\}\) | \(\{(0.5e^{i0.7\pi })\}\) |
c | d | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
e | d | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) | \(\varnothing \) |
Application
Name of predator | Name of prey | Like to eat\((\%)\) | Tasty to eat\((\%)\) |
---|---|---|---|
Killer whale | Sea otter | 60 | 70 |
Sea otter | Sea urchin | 60 | 45 |
Sea otter | Starfish | 70 | 55 |
Starfish | Sea urchin | 80 | 40 |
Starfish | Snail | 70 | 40 |
Snail | Seaweed | 70 | 20 |
Sea urchin | Seaweed | 70 | 20 |
\(y \in Z\) | \(\aleph ^{p}(y)\) |
---|---|
Killer whale | \(\phi \) |
Sea otter | \(\left\{ \left( \mathrm{killer whale},0.6e^{i1.4\pi }\right) \right\} \) |
Sea urchin | {(sea otter, \(0.6e^{i0.9\pi }\)), (starfish,\(0.8e^{i0.8\pi }\))} |
Starfish | { (sea otter,\(0.7e^{i1.1\pi })\)} |
Snail | {(starfish,\(0.7e^{i0.4\pi })\)} |
Seaweed | {(sea urchin,\(0.7e^{i0.4\pi })\),(snail,\(0.7e^{i0.4\pi }\))} |
Method | Ecosystem |
---|---|
Step 1 | Assign the MVs for the set of n species in the food web. \(\overrightarrow{\Omega }=(\mathbb {\widehat{P}},\overrightarrow{\mathbb {\widehat{Q}}})\)(say). |
Step 2 | If for any two nodes \(y_{i}\) and \(y_{j},\) \(\xi _{\mathbb {\widehat{Q}}}(y_{i},y_{j})>0,\) then |
\((y_{j},\xi _{\mathbb {\widehat{Q}}}(y_{i},y_{j})e^{i\varphi _{\mathbb {\widehat{Q}}}(y_{i},y_{j})})\in \aleph ^{p}(y_{i}).\) | |
Step 3 | Find out the CFON \(\aleph ^{p}(y_{i})\) for all vertices \(y_{i},y_{j}\). |
Step 4 | calculate the \(\aleph ^{p}(y_{i})\cap \aleph ^{p}(y_{j}).\) |
Step 5 | calculate \(\hbar (\aleph ^{p}(y_{i})\cap \aleph ^{p}(y_{j})).\) |
Step 6 | If \(\aleph ^{p}(y_{i})\cap \aleph ^{p}(y_{j})\ne \varnothing \) then draw an edge \((y_{i},y_{j}).\) |
Step 7 | Repeat the Step 6 for all the disjoint nodes of the digraph. |
Step 8 | Calculate the MVs using |
\(\xi _{\mathbb {\widehat{Q}}}(y_{i},y_{j})=(\xi _{\mathbb {\widehat{P}}}(y_{i})\wedge \xi _{\mathbb {\widehat{Q}}}y_{j})\times {\hbar }_{\xi }(\aleph ^{p}(y_{i})\cap \aleph ^{p}(y_{j}))\) and | |
\(\varphi _{\mathbb {\widehat{Q}}}(y_{i},y_{j})=2\pi \left[ \left( \frac{\varphi _{\mathbb {\widehat{P}}}(y_{i})}{2\pi }\wedge \frac{\varphi _{\mathbb {\widehat{P}}}( y_{j})}{2\pi }\right) \times \frac{{\hbar }_{\varphi }(\aleph ^{p}(y_{i})\cap \aleph ^{p}(y_{j}))}{2\pi }\right] .\) |
Comparative analysis
Further discussion
Method | Whether have ability | Whether have ability | Whether have the characteristics |
---|---|---|---|
to handle periodic problems | to represent 2-D information | of generalization | |
Samanta et al. [20] | \(\times \) | \(\times \) | \(\times \) |
Sahoo and Pal [18] | \(\times \) | \(\times \) | \(\times \) |
Sarwar [21] | \(\times \) | \(\times \) | \(\times \) |
Habib et al. [9] | \(\times \) | \(\times \) | \(\checkmark \) |
The proposed CFCGs | \(\checkmark \) | \(\checkmark \) | \(\checkmark \) |