Introduction
Research gap and motivation
Main contributions
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The edge F-index is proposed for fuzzy graphs here. Bounds of this index are calculated for FGs. The FG has been investigated for a given set of vertices as having maximum edge F-index. For an isomorphic FGs, it is shown that the value of this index is the same.
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Some relations of this index with the second Zagreb index and hyper-Zagreb index are established.
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Bounds of this index for some FG operations are determined.
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An application of the index in mathematical chemistry is studied. For this, 18 octane isomers and 67 alkanes are considered and analyzed the correlation between this index with some properties of the octane isomers and alkanes. From the correlation coefficient value, we have obtained this index is highly correlated with enthalpy of vaporization, standard enthalpy of vaporization, entropy, acentric factor and heat of vaporization and less correlated with heat capacity for octane isomers. Also, this index is correlated with critical pressure, critical temperature, molar refraction, molar volume and boiling point and is less correlated with surface tension and heat of vaporization for alkanes. But, this index is inadequate to determine the melting point of alkanes.
Paper organization
Preliminaries
Edge F-index for a fuzzy graph
Relation of edge F-index with other topological indices for fuzzy graph
Edge F-index for fuzzy graph operations
Application of edge F-index in mathematical chemistry
QSPR analysis of edge F-index for octane isomers
Octane isomers | HC | EV | SEV | HV | AF | E | EFI |
---|---|---|---|---|---|---|---|
n-octane | 24.64 | 73.19 | 9.915 | 41.48 | 0.397898 | 111.67 | 5.12 |
2-Methyl heptane | 24.8 | 70.3 | 9.484 | 39.68 | 0.377916 | 109.84 | 6.3488 |
3-Methyl heptane | 25.6 | 70.91 | 9.521 | 39.83 | 0.371002 | 111.26 | 6.3488 |
4-Methyl heptane | 25.6 | 70.91 | 9.483 | 39.69 | 0.371504 | 109.32 | 6.3488 |
3-Ethyl hexane | 25.74 | 71.7 | 9.476 | 39.4 | 0.362472 | 109.43 | 6.3488 |
2,2-Dimethyl hexane | 25.6 | 67.7 | 8.915 | 37.29 | 0.339426 | 103.42 | 9.4208 |
2,3-Dimethyl hexane | 26.6 | 70.2 | 9.272 | 38.79 | 0.348247 | 108.02 | 7.5776 |
2,4-Dimethyl hexane | 25.8 | 68.5 | 9.029 | 37.76 | 0.344223 | 106.98 | 7.5776 |
2,5-Dimethyl hexane | 25 | 68.6 | 9.051 | 37.86 | 0.35683 | 105.72 | 7.5776 |
3,3-Dimethyl hexane | 27.2 | 68.5 | 8.973 | 37.93 | 0.322569 | 104.74 | 9.4208 |
3,4-Dimethyl hexane | 27.4 | 70.2 | 9.316 | 39.02 | 0.340345 | 106.59 | 7.5776 |
2-Methyl-3-ethyl pentane | 27.4 | 69.7 | 9.209 | 38.52 | 0.332433 | 106.06 | 7.5776 |
3-Methyl-3-ethyl pentane | 28 | 69.3 | 9.081 | 37.99 | 0.306899 | 101.48 | 9.4208 |
2,2,3-Trimethyl pentane | 28.2 | 67.3 | 8.826 | 36.91 | 0.300816 | 101.31 | 10.6496 |
2,2,4-Trimethyl pentane | 25.5 | 64.87 | 8.402 | 35.13 | 0.30537 | 104.09 | 10.6496 |
2,3,3-Trimethyl pentane | 29 | 68.1 | 8.897 | 37.22 | 0.293177 | 102.06 | 9.9328 |
2,3,4-Trimethyl pentane | 27.6 | 68.37 | 9.014 | 37.61 | 0.317422 | 102.39 | 8.8064 |
2,2,3,3-Tetramethyl butane | 66.2 | 8.41 | 0.255294 | 93.06 | 13.7216 |
QSPR analysis of edge F-index for alkanes
Alkanes | MP | ST | CP | HV | MR | MV | BP | EFI |
---|---|---|---|---|---|---|---|---|
Butane | \(-\)138.35 | 37.47 | \(-\)0.05 | 1.8432 | ||||
2-Methyl propane | \(-\)159.6 | 36 | \(-\)11.73 | 3.072 | ||||
Pentane | \(-\)129.72 | 16 | 33.31 | 26.42 | 25.2656 | 115.205 | 36.074 | 2.6624 |
2-Methyl butane | \(-\)159.9 | 15 | 32.9 | 24.59 | 25.2923 | 116.426 | 27.852 | 3.8912 |
2,2-Dimethyl propane | \(-\)16.55 | 31.57 | 21.78 | 25.7243 | 112.074 | 9.503 | 6.9632 | |
Hexane | \(-\)95.35 | 18.42 | 29.92 | 31.55 | 29.9066 | 130.688 | 68.74 | 3.4816 |
2-Methyl pentane | \(-\)153.67 | 17.38 | 29.95 | 29.86 | 29.945 | 131.933 | 60.271 | 4.7104 |
3-Methyl pentane |
\(-118\)
| 18.12 | 30.83 | 30.27 | 29.8016 | 129.717 | 63.282 | 4.7104 |
2,2-Methyl butane | \(-\)99.87 | 16.3 | 30.67 | 27.69 | 29.9347 | 132.744 | 4.741 | 7.7824 |
2,3-Dimethyl butane | \(-\)128.54 | 17.37 | 30.99 | 29.12 | 29.8104 | 130.24 | 57.988 | 5.9392 |
Heptane | \(-\)90.61 | 20.26 | 27.01 | 36.55 | 34.5504 | 146.54 | 98.427 | 4.3008 |
2-Methyl hexane | \(-\)118.28 | 19.29 | 27.2 | 34.8 | 34.5908 | 147.656 | 90.052 | 5.5296 |
3-Methyl hexane | \(-\)119.4 | 19.79 | 28.1 | 35.08 | 34.4597 | 145.821 | 91.85 | 5.5296 |
3-Ethyl pentane | \(-\)118.6 | 20.44 | 28.6 | 35.22 | 34.2827 | 143.517 | 93.475 | 5.5296 |
2,2-Dimethyl pentane | \(-\)123.81 | 18.02 | 28.4 | 32.43 | 34.6166 | 148.695 | 79.197 | 8.6016 |
2,3-Dimethyl pentane | \(-\)119.1 | 19.96 | 29.2 | 34.24 | 34.3237 | 144.153 | 89.784 | 6.7584 |
2,4-Dimethyl pentane | \(-\)119.24 | 18.15 | 27.4 | 32.88 | 34.6192 | 148.949 | 80.5 | 6.7584 |
3,3-Dimethyl pentane | \(-\)134.46 | 19.59 | 30 | 33.02 | 34.3323 | 144.53 | 86.064 | 8.6016 |
n-Octane | \(-\)56.79 | 21.76 | 24.64 | 41.48 | 39.1922 | 162.592 | 125.665 | 5.12 |
2-Methyl heptane | \(-\)109.04 | 20.6 | 24.8 | 39.68 | 39.2316 | 163.663 | 117.647 | 6.3488 |
3-Methyl heptane | \(-\)120.5 | 21.17 | 25.6 | 39.83 | 39.1001 | 161.832 | 118.925 | 6.3488 |
4-Methyl heptane | \(-\)120.95 | 21 | 25.6 | 39.67 | 39.1174 | 162.105 | 117.709 | 6.3488 |
3-Ethyl hexane | 21.51 | 25.74 | 39.4 | 38.94 | 160.07 | 118.53 | 6.3488 | |
2,2-Dimethyl hexane | \(-\)121.18 | 19.6 | 25.6 | 37.29 | 39.25 | 164.28 | 10.84 | 9.4208 |
2,3-Dimethyl hexane | 20.99 | 26.6 | 38.79 | 38.98 | 160.39 | 115.607 | 7.5776 | |
2,4-Dimethyl hexane | \(-\)137.5 | 20.05 | 25.8 | 37.76 | 39.13 | 163.09 | 109.42 | 7.5776 |
2,5-Dimethyl hexane | \(-\)91.2 | 19.73 | 25 | 37.86 | 39.25 | 164.69 | 109.1 | 7.5776 |
3,3-Dimethyl hexane | \(-\)126.1 | 20.63 | 27.2 | 37.93 | 39 | 160.87 | 111.96 | 9.4208 |
3,4-Dimethyl hexane | 21.64 | 27.4 | 39.02 | 38.84 | 158.81 | 117.72 | 7.5776 | |
2-Methyl-3-ethyl pentane | \(-\)114.96 | 21.52 | 27.4 | 38.52 | 38.83 | 158.79 | 115.65 | 7.5776 |
3-Methyl-3-ethyl pentane | \(-\)90.87 | 21.99 | 28 | 37.99 | 38.71 | 157.02 | 118.25 | 9.4208 |
2,2,3-Trimethyl pentane | \(-\)112.27 | 20.67 | 28.2 | 36.91 | 38.92 | 159.52 | 109.84 | 10.6496 |
2,2,4-Trimethyl pentane | \(-\)107.38 | 18.77 | 25.5 | 35.13 | 39.26 | 165.08 | 99.23 | 10.6496 |
2,3,3-Trimethyl pentane | \(-\)100.7 | 21.56 | 29 | 37.22 | 38.76 | 157.29 | 114.76 | 9.9328 |
2,3,4-Trimethyl pentane | \(-\)109.21 | 21.14 | 27.6 | 37.61 | 38.86 | 158.85 | 113.46 | 8.8064 |
Nonane | \(-\)53.52 | 22.92 | 22.74 | 46.44 | 43.84 | 178.71 | 150.76 | 5.9392 |
2-Methyl octane | \(-\)80.4 | 21.88 | 23.6 | 44.65 | 43.87 | 179.77 | 143.26 | 7.168 |
3-Methyl octane | \(-\)107.64 | 22.34 | 23.7 | 44.75 | 43.72 | 177.5 | 144.18 | 7.168 |
4-Methyl octane | \(-\)113.2 | 22.34 | 23.06 | 44.75 | 43.76 | 178.15 | 142.48 | 7.168 |
3-Ethyl heptane | \(-\)114.9 | 22.81 | 23.98 | 44.81 | 43.64 | 176.41 | 143 | 7.168 |
4-Ethyl heptane | 22.81 | 23.98 | 44.81 | 43.69 | 175.68 | 141.2 | 7.168 | |
2,2-Dimethyl heptane | -113 | 20.8 | 22.8 | 42.28 | 43.91 | 180.5 | 132.69 | 10.24 |
2,3-Dimethyl heptane | -116 | 22.34 | 23.79 | 43.79 | 43.63 | 176.65 | 140.5 | 8.3968 |
2,4-Dimethyl heptane | 23.3 | 22.7 | 42.87 | 43.73 | 179.12 | 133.5 | 8.3968 | |
2,5-Dimethyl heptane | 21.3 | 22.7 | 43.87 | 43.84 | 179.37 | 136 | 8.3968 | |
2,6-Dimethyl heptane | \(-\)102.9 | 20.83 | 23.7 | 42.82 | 43.92 | 180.91 | 135.21 | 8.3968 |
3,3-Dimethyl heptane | 22.01 | 24.19 | 42.66 | 43.687 | 176.897 | 137.3 | 10.24 | |
3,4-Dimethyl heptane | 22.8 | 24.77 | 43.84 | 43.5473 | 175.349 | 140.6 | 8.3968 | |
3,5-Dimethyl heptane | 21.77 | 23.59 | 42.98 | 43.6379 | 177.386 | 136 | 8.3968 | |
4,4-Dimethyl heptane | 22.01 | 24.18 | 42.66 | 43.6022 | 176.897 | 135.2 | 10.24 | |
3-Ethyl-2-methyl hexane | 22.8 | 24.77 | 43.84 | 43.655 | 175.445 | 138 | 8.3968 | |
4-Ethyl-2-methyl hexane | 21.77 | 25.56 | 42.98 | 43.6472 | 177.386 | 133.8 | 8.3968 | |
3-Ethyl-3-methyl hexane | 23.22 | 25.66 | 44.04 | 43.268 | 173.077 | 140.6 | 10.24 | |
2,2,4-Trimethyl hexane |
\(-120\)
| 20.51 | 23.39 | 40.57 | 43.7638 | 179.22 | 126.54 | 10.1376 |
2,2,5-Trimethyl hexane | \(-\)105.78 | 20.04 | 22.41 | 40.17 | 43.9356 | 181.346 | 124.084 | 11.4688 |
2,3,3-Trimethyl hexane | \(-\)116.8 | 22.41 | 25.56 | 42.23 | 43.4347 | 173.78 | 137.68 | 11.4688 |
2,3,4-Trimethyl hexane | 22.8 | 25.46 | 42.93 | 43.4917 | 173.498 | 139 | 9.6256 | |
2,3,5-Trimethyl hexane | \(-\)127.8 | 21.27 | 23.49 | 41.42 | 43.6474 | 177.656 | 131.34 | 9.6256 |
3,3,4-Trimethyl hexane | \(-\)101.2 | 23.27 | 26.45 | 42.28 | 43.3407 | 172.055 | 140.46 | 10.4448 |
3,3-Diethyl pentane | \(-\)33.11 | 23.75 | 26.94 | 43.36 | 43.1134 | 170.185 | 146.168 | 10.24 |
2,2-Dimethyl-3-ethyl pentane | \(-\)99.2 | 22.38 | 25.96 | 42.02 | 43.4571 | 174.537 | 133.83 | 10.1376 |
2,3-Dimethyl-3-ethyl pentane | 23.87 | 26.94 | 42.55 | 42.9542 | 170.093 | 142 | 11.4688 | |
2,4-Dimethyl-3-ethyl pentane | \(-\)122.2 | 22.8 | 25.46 | 42.93 | 43.4037 | 173.804 | 136.73 | 9.6256 |
2,2,3,3-Tetramethyl pentane |
\(-99\)
| 23.38 | 27.04 | 41 | 43.2147 | 169.495 | 140.274 | 14.5408 |
2,2,3,4-Tetramethyl pentane |
\(-121.09\)
| 21.8 | 25.66 | 41 | 43.4359 | 173.557 | 133.016 | 12.6976 |
2,2,4,4-Tetramethyl pentane | \(-\)66.54 | 20.37 | 24.58 | 38.1 | 43.8747 | 178.256 | 122.284 | 14.5408 |
2,3,3,4-Tetramethyl pentane | \(-\)102.12 | 23.31 | 26.85 | 41.75 | 43.2016 | 169.928 | 141.551 | 12.6976 |