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Energy and Laplacian Energy-Like Invariant of Unicyclic Molecular Graphs

Masood Ur Rehman*, Muhammad Riaz

University of Science and Technology of China, School of Mathematical Sciences, Hefei, An-hui, 230026, P.R. China

*Corresponding Author:
Masood Ur Rehman
University of Science and Technology of China
School of Mathematical Sciences
Hefei, An-hui, 230026, P.R. China
 
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Abstract

Let Un be the set of unicyclic molecular graphs with 3 ≤ n ≤ 8 vertices. We show that the cycle Cn has maximal Laplacian-energy-like invariant (LEL) in Un. The authors partially proving that the conjecture hold for any unicyclic molecular graph in Un, where 3 ≤ n ≤ 8 Moreover, we show that Cn has maximal energy (E) in Un for 3 ≤ n≤7, but for n=8 this is not true.

Keywords:

Molecular graphs, Laplacian energy-like invariant, Energy

Introduction

The total π-electron energy E, as calculated within the Huckel molecular orbital (HMO) model, is one of the most thoroughly studied quantum-chemical characteristics of large polycyclic conjugated molecules. Details on the theory and applications of E can be found in the literature [1-3] and in the references cited therein. It was recognized a long time ago that the various π-electron descriptors of HMO model, including E, can be calculated from the eigenvalues λ1, λ2,..., λn of the underline molecular graph [4,5]. In particular, in the case of alternant hydrocarbons.

images (1)

Where, as usual [1,2,4,5] E is expressed in the units of the HMO carbon-carbon resonance integral β. Formula (1) served as a motivation for the definition of the so-called graph energy. Namely, whereas within the HMO model E is meaningful only in the case of a restricted class of molecular graphs [5], the right-hand side of (1.1) is a well-defined quantity for all graphs. In view of this, the energy of a graph (also denoted by E) is defined as the sum of the absolute values of all eigenvalues of this graph, and this definition extends to all graphs. This seemingly insignificant change in the interpretation of Equation. (1) resulted in a great expansion of research in this area and has advanced the theory of total π-electron energy greatly; for details see the reviews [1,6] and some of the most recent publications dealing with graph energy [7-13].

By equation (1), the graph energy is defined in terms of the graph eigenvalues λ1, λ2

,..., λn. Recall that these are just the eigenvalues of the adjacency matrix [14]. Motivated by the success of the graph energy concept, and in order to extend it to the Laplacian eigenvalues, the Laplacian energy LE(G) was put forward, defined as

images (2)

Where G is a graph with n vertices and m edges, and μ1, μ2,….., μn are its Laplacian eigenvalues [15]. The Laplacian energy has two major drawbacks: Namely, neither LE (G1G2)=LE (G1) + LE (G2) holds in the general case, for G1G2 being the graph consisting of two disconnected components G1 and G2, nor is the condition LE (GK1)=LE (G), where K1 is the graph with single vertex, satisfied. In order to overcome these difficulties, Liu and Liu invented the Laplacian energy-like invariant LEL (G), defined as [16].

images (3)

Indeed, the relations LEL (G1 [ G2)=LEL(G1) + LEL (G2) and LE(G ] K1)=LE(G) are generally vaild.

The theory of LEL is nowadays well developed; details and further references can be found in the review [17]. In particular, numerous correlations between LEL and physico-chemical properties of alkanes were reported [18]. It was shown that, in spite of its name, LEL resembles more the total π-electron energy than the Laplacian energy LE [19]. It is known that the main parameters determining the value of the total π-electron energy are n (= the number of carbon atoms, i.e, the number of vertices of the molecular graph) and m (= the number of carbon- carbon bonds, i.e, the number of edges of the molecular graph) [1,20].

In [21], Stevanovi´c studied the LEL of trees (trees are connected acyclic graphs). In that paper he conjectured the following conjecture as well

Conjecture 1.1 Among unicyclic molecular graphs on n vertices, the cycle Cn has maximal Laplacian energy-like invariant.

The main purpose of the present paper is to give the partial proof of the conjecture (1.1). That is we will prove that the conjecture 1.1 is true for 3 ≤ n ≤ 8. Moreover, we will show that among molecular unicyclic graphs the cycle Cn has maximal energy E for 3 ≤ n ≤ 7 but for n=8 it is not true. Where n is the number of vertices in the unicyclic molecular graph.

For unexplained terminology see the following subsection

Definitions: Let G=(V, E) be a simple connected molecular graph with vertex set V={v1, v2,..., vn} and edge set E={e1, e2,..., en}. Its adjacency matrix A(G)=(aij) is defined as n × n matrix (aij), where aij=1 if vi is adjacent to vj; and aij=0, otherwise. Denote by d(vi) or dG(vi) the degree of the vertex vi (number of adjacent vertices to vi). The matrix L(G)=D(G) − A(G) is called the Laplacian matrix of graph G, where D(G)=diag(dv1, dv2,..., dvn ) denotes the diagonal matrix of vertex degrees of G.

Definition: A unicyclic molecular graph is a connected graph G containing exactly one cycle.

Unicyclic Molecular Graphs

In this section, we will give all unicyclic molecular graphs with at most 8 vertices.

Lemma

There are exactly 143 unicyclic molecular graphs with 3 ≤ n ≤ 8 vertices.

Proof. From [22, p. 213-215] one can find all unicyclic molecular graphs for 3 ≤ n ≤ 8, where n is the number of vertices in the unicyclic molecular graph G

On (Figure 1), we give all the diagrams of the 54 unicyclic molecular graphs with at most 7 vertices. On (Figure 2), we give all the diagrams of the 89 unicyclic molecular graphs with exactly 8 vertices. We denote all these unicyclic molecular graphs by Ui for i=1,..., 143. Clearly, on (Figure 1), U1 is a cycle C3, U3 is a cycle C4, U8 is a cycle C5, U21 is a cycle C6 and U54 is a cycle C7. On Figure 2, U143 is a cycle C8.

Partial Proof Of The Conjecture

From, (Figure 1 and 2), we have all unicyclic molecular graphs Ui for i=1,..., 143 with at most 8 vertices. In the first column of the following tables (Tables 1-3), we give the name Ui for i=1,..., 143, in the second column the number of vertices n, in the third column the Laplacian spectrum and in the fourth column we give Laplacian energy-like invariant LEL. By direct calculation (one can do this exercise by computer, by use of suitable mathematical softwares, for example Matlab or athematica) we find the Laplacian spectrum of each unicyclic molecular graph Ui and the Laplacian energy-like invariant LEL.

der-chemica-sinica-molecular-graphs

Figure 1: Diagrams of the 54 unicyclic molecular graphs with at most 7 vertices.

der-chemica-sinica-molecular-graphs

Figure 2: Diagrams of the 89 unicyclic molecular graphs with exactly 8 vertices.

Now, if n=3, then we have only one unicyclic molecular graph U1=C3, with LEL ≈ 3.4146, for n=4, U3=C4 has the maximal LEL among all unicyclic molecular graphs with 4 (Tables 1 and 2) vertices, for n=5, U8=C5 has the maximal LEL among all unicyclic molecular graphs with 5 (Table 3) vertices, for n=6, U21=C6 has the maximal LEL among all the unicyclic molecular graphs with 6 vertices, for n=7, U54=C7 has the maximal LEL among all unicyclic molecular graphs with 7 vertices and for n=8, U143=C8 has the maximal LEL among all unicyclic molecular graphs with 8 vertices. Consequently, we can see that the conjecture 1.1 is true for 3 ≤ n ≤ 8.

# n Laplacian spectrum LEL
U1 3 0, 3, 3 3.4641
U2 4 0, 1, 3, 4 4.7321
U3 4 0, 2, 2, 4 4.8284
U4 5 0, 0.5188, 2.3111, 3, 4.1701 6.0146
U5 5 0, 0.6972, 1.3820, 3.6180, 4.3028 5.987
U6 5 0, 1, 1, 3, 5 5.9681
U7 5 0, 0.8299, 2, 2.6889, 4.4812 6.0819
U8 5 0, 1.3820, 1.3820, 3.6180, 3.6180 6.1554
U9 6 0, 0.3249, 1.4608, 3, 3, 4.2143 7.2956
U10 6 0, 0.6571, 1, 2.5293, 3, 4.8136 7.327
U11 6 0, 0.7639, 1, 2, 3, 5.2361 7.3085
U12 6 0, 0.4384, 1, 3, 3, 4.5616 7.262
U13 6 0, 0.4859, 1, 2.4280, 3, 5.0861 7.2426
U14 6 0, 1, 1, 1, 1, 3, 6 7.1815
U15 6 0, 0.6314, 1, 1.4738, 3.7877, 5.1071 7.2147
U16 6 0, 0.4131, 1.1369, 2.3595, 3.6977, 4.3928 7.2639
U17 6 0, 0.6972, 0.6972, 2, 4.3028, 4.3028 7.2328
U18 6 0, 0.4384, 2, 2, 3, 4.5616 7.3584
U19 6 0, 0.5858, 1.2679, 2, 3.4142, 4.7321 7.3287
U20 6 0, 0.6972, 1.3820, 2, 3.6180, 4.3028 7.4012
U21 6 0, 1, 1, 3, 3, 4 7.4641
U22 7 0, 1, 1, 1, 1, 3, 7 8.3778
U23 7 0, 0.5961, 1, 1, 1.5196, 3.8273, 6.0570 8.4222
U24 7 0, 0.5505, 1, 1, 1.5858, 4.4142, 5.4495 8.4367
U25 7 0, 0.6086, 0.6972, 1, 2.2271, 4.3028, 5.1642 8.4543
U26 7 0, 0.4659, 1, 1, 2.4827,3, 6.0514 8.4502
U27 7 0, 0.7269, 1, 1, 2, 3.1404, 6.1326 8.5153
U28 7 0, 0.3983, 1, 1, 3, 3.3399, 5.2618 8.4846
U29 7 0, 0.3983, 1, 1, 3, 3.3399, 5.2618 8.4846
U30 7 0, 0.4116, 0.7530, 1.4064, 2.4450, 3.8019, 5.1819 8.4851
U31 7 0, 0.3679, 1, 1.1879, 2.3732, 3.9464, 5.1228 8.4869
U32 7 0, 0.5858, 1, 1, 2.5858, 3.4142, 5.4142 8.548
U33 7 0, 0.5140, 1, 1.3364, 2, 3.8360, 5.3136 8.5514
U34 7 0, 0.3820, 0.6972, 1.5858, 2.6180, 4.3028, 4.4142 8.5057
U35 7 0, 0.3403, 1, 1.1451, 3, 3.8549, 4.6567 8.5075
U36 7 0, 0.5858, 0.6837, 1.4206, 2.8654, 3.4142, 5.0303 8.5675
U37 7 0, 0.2955, 1, 1.4911, 3, 3.1169, 5.09665 8.5198
U38 7 0, 0.3820, 0.6086, 2.2271, 2.6180, 3, 5.1642 8.5131
U39 7 0, 0.4330, 0.8510, 2, 2.3024, 3.1129, 5.3006 8.5787
U40 7 0, 0.6086, 1, 1.3820, 2.2271, 3.6180, 5.1642 8.6227
U41 7 0, 0.3004, 0.7530, 2.2391, 2.4450, 3.8019, 4.4605 8.5377
U42 7 0, 0.2679, 1, 1.5858, 3, 3.7321, 4.4142 8.5418
U43 7 0, 0.3217, 0.6802, 2.1397, 3, 3.2297, 4.6287 8.5353
U44 7 0, 0.2679, 1, 1.5858, 3, 3.7321, 4.4142 8.5418
U45 7 0, 0.3820, 0.8851, 2, 2.6180, 3.2541, 4.8608 8.5997
U46 7 0, 0.3588, 1, 2, 2.2763, 3.5892, 4.7757 8.6018
U47 7 0, 0.3588, 1, 2, 2.2763, 3.5892, 4.7757 8.6018
U48 7 0, 0.5188, 1, 1.5858, 2.3111, 4.1701, 4.4142 8.6429
    6  

Table 1: Partial proof of the conjecture.

# n Laplacian spectrum LEL
U49 7 0, 0.6228, 0.7530, 1.7261, 2.4450, 3.8019, 4.6511 8.6409
U50 7 0, 0.2254, 1, 2.1859, 3, 3.3604, 4.2283 8.5747
U51 7 0, 0.2765, 1.3323, 2, 2.5219, 3.2920, 4.5772 8.6362
U52 7 0, 0.3820, 1.3820, 1.5858 ,2.6180, 3.6180, 4.4142 8.6741
U53 7 0, 0.5858, 1, 1.5858, 3, 3.4142, 4.4142 8.7055
U54 7 0, 0.7530, 0.7530, 2.4450, 2.4450, 3.8019, 3.8019 8.7625
U55 8 0, 1, 1, 1, 1, 1, 3, 8 9.5605
U56 8 0, 1, 1, 1, 1.5468, 7.0362 9.6142
U57 8 0, 0.5069, 1, 1, 1, 1.6400, 4.6654, 6.1877 9.6401
U58 8 0, 0.5607, 0.9672, 1, 1, 2.3389, 4.3028, 6.1004 9.6573
U59 8 0, 0.5505, 0.6571, 1, 1, 2.5293, 4.8136, 5.4495 9.9714
U60 8 0, 0.7029, 1, 1, 1, 2, 3.2132, 7.0839 9.7067
U61 8 0, 0.4592, 1, 1, 1, 2.5135, 3, 7.0340 9.6423
U62 8 0, 0.3738, 1, 1, 1, 3, 3.4849, 6.1413 9.6884
U63 8 0, 0.5449, 1, 1, 1, 2.5987, 3.6291, 6.2273 9.7507
U64 8 0, 0.4746, 1, 1, 1.3691, 2, 4, 6.1563 9.7544
U65 8 0, 0.3738, 1, 1, 1, 3, 3.4849, 6.1413 9.6884
U66 8 0, 0.3417, 1, 1, 1.2176, 2.3795, 4, 6.0612 9.6925
U67 8 0, 0.4103, 0.6758, 1, 1.4853, 2.4907, 3.8322, 6.1057 9.6881
U68 8 0, 0.3542, 1, 1, 1, 3, 4, 5.6458 9.7033
U69 8 0, 0.5107, 1, 1, 1, 2.7108, 4, 5.7785 9.7649
U70 8 0, 0.4384, 1, 1, 1.4384, 2, 4.5616, 5.5616 9.7698
U71 8 0, 0.3676, 0.7223, 1, 1.5047, 2.4500, 4.4669, 5.4885 9.7044
U72 8 0, 0.3568, 0.6365, 1, 1.6887, 2.7571, 4.3873, 5.1735 9.7242
U73 8 0, 0.5858, 0.5858, 1, 1.4384, 3.4142, 3.4142, 5.5616 9.7839
U74 8 0, 0.5066, 0.6743, 1, 1.4986, 2.8999, 3.9168, 5.5038 9.7851
U75 8 0, 0.3339, 0.7460, 1, 1.4123, 3.3136, 3.8587, 5.3356 9.7245
U76 8 0, 0.3030, 1, 1, 1.1479, 3.2427, 4, 5.2863 9.7218
U77 8 0, 0.2967, 1, 1, 1.2048, 3, 4.3310, 5.1675 9.7287
U78 8 0, 0.3820, 0.6972, 0.7639, 2, 2.6180, 4.3028, 5.2361 9.7219
U79 8 0, 0.3065, 0.6972, 1, 1.6703, 3.3297, 4.3028, 4.6935 9.7465
U80 8 0, 0.5858, 0.5858, 0.7639, 2, 3.4142, 3.4142, 5.2361 9.8027
U81 8 0, 0.3820, 0.5607, 1, 2.3389, 2.6180, 3, 6.1004 9.7162
U82 8 0, 0.5607, 1, 1, 1.3820, 2.3389, 3.6180, 6.1004 9.8257
U83 8 0, 0.4284, 0.7828, 1, 2, 2.4204, 3.1905, 6.1779 9.781
U84 8 0, 0.2774, 1, 1, 1.5068, 3, 3.1610, 6.0548 9.7248
U85 8 0, 0.2888, 0.6742, 1, 2.1694, 3, 3.5857, 5.2819 9.7553
U86 8 0, 0.5447, 0.7347, 1, 1.7635, 2.7242, 3.9063, 5.3266 9.858
U87 8 0, 0.4484, 1, 1, 1.6280, 2.4815, 4.2659, 5.1762 9.8614
U88 8 0, 0.3479, 0.8495, 1, 2, 2.7627, 3.6076, 5.4323 9.818
U89 8 0, 0.3187, 1, 1, 2, 2.3579, 4, 5.3234 9.8215
U90 8 0, 0.3820, 0.7254, 1, 2.3000, 2.6180, 3.5096, 5.4651 8.5418
U91 8 0, 0.3581, 0.6918, 1.2843, 2, 2.4091, 3.8877, 5.3689 9.8186
U91 8 0, 0.3479, 0.8495, 1, 2, 2.7627, 3.6076, 5.4323 9.818
U93 8 0, 0.3187, 1, 1, 2, 2.3579, 4, 5.3234 9.8215
U94 8 0, 0.3187, 0.5858, 1, 2.3579, 3, 3.4142, 5.3234 9.7525
U95 8 0, 0.2384, 1, 1, 1.6367, 3, 4, 5.1249 9.7635
U96 8 0, 0.2955, 0.5979, 1449, 2.3295, 2.4734, 3.9635, 5.1952 9.756
    7  

Table 2 : Partial proof of the conjecture.

# n Laplacian spectrum   LEL
U97 8 0, 0.3187, 0.5858, 1, 2.3579, 3, 3.4142, 5.3234   9.7525
U98 8 0, 0.2384, 1, 1, 1.6367, 3, 4, 5.1249   9.7635
U99 8 0, 0.2593, 0.7150, 1.3232, 1.5891, 3.1143, 3.8086, 5.1905   9.7603
U100 8 0, 0.3820, 0.4280, 1.2285 ,2.2799, 2.6180, 3.8123, 5.2513   9.7527
U101 8 0, 0.2384, 1, 1, 1.6367, 3, 4, 5.1249   9.7635
U102 8 0, 0.3004, 0.4915, 1.3204, 2.2391, 2.8258, 4.3623, 4.4605   9.7762
U103 8 0, 0.5188, 0.6571, 1, 2.3111, 2.5293, 4.1701, 4.8136 9.8776
U104 8 0, 0.4915, 0.6228, 1.3204, 1.7261, 2.8258, 4.3623, 4.6511   9.8794
U105 8 0, 0.3074, 0.8828, 1, 2.2699, 2.7125, 3.8417, 4.9857 9.8405
U106 8 0, 0.2907, 1, 1, 2, 2.8061, 4, 4.9032   9.8428
U107 8 0, 0.3636, 0.5858, 1.3478, 2, 3.2222, 3.4142, 5.0664 9.8372
U108 8 0, 0.3432, 0.6639, 1.1805, 2.2491, 2.9045, 3.5994, 5.0594   9.8376
U109 8 0, 0.2679, 0.6571, 1, 2.5293, 3, 3.7321, 4.8136   9.7765
U110 8 0, 0.2588, 0.6436, 1.1385, 2.1603, 3.1943, 3.8943, 4.7103   9.7788
U111 8 0, 0.2183, 1, 1, 1.7127, 3.5524, 4, 4.5166   9.7859
U112 8 0, 0.2509, 0.7287, 1, 2.3349, 3, 4, 4.6855   9.7792
U113 8 0, 0.2434, 0.6972, 1.1798, 2, 3.1386, 4.3028, 4.4383 9.7814
U114 8 0, 0.2023, 1, 1, 2.2472, 3, 3.4527, 5.0979   9.7969
U115 8 0, 0.4965, 1, 1, 1.7356, 3, 3.5767, 5.1912   9.9237
U116 8 0, 0.3820, 0.7639, 1.3820, 2, 2.6180, 3.6180, 5.2361 9.8903
U117 8 0, 0.2652, 0.8350, 1.4524, 2, 2.8415, 3.2984, 5.3075 9.8538
U118 8 0, 0.3820, 0.4711, 2, 2, 2.6180, 3.1674, 5.3615   9.8461
U119 8 0, 0.2538, 0.5472, 1.4689, 2.4066, 3, 3.1504, 5.1732 9.7883
U120 8 0, 0.5858, 0.5858, 1.2679, 2, 3.4142, 3.4142, 4.7321 9.9418
U121 8 0, 0.4679, 0.7369, 1.4843, 1.6527, 3.1826, 3.8794, 4.5962   9.9438
U122 8 0, 0.4384, 1, 1, 2, 3, 4, 4.5616   9.9442
U123 8 0, 0.3095, 1, 1.3820, 1.6703, 3.3297, 3.6180, 4.6935 9.9149
U124 8 0, 0.3547, 0.7089, 1.5498, 2, 2.8407, 3.8349, 4.711 9.9109
U125 8 0, 0.3249, 0.8299, 1.4608, 2, 2.6889, 4.2143, 4.4812 9.9134
U126 8 0, 0.2679, 0.6571, 2, 2, 2.5293, 3.7321, 4.8136   9.8729
U127 8 0, 0.2243, 1, 1.4108, 2, 2.7237, 4, 4.6412   9.8803
U128 8 0, 0.2442, 0.8455, 1.3465, 2.4678, 2.7742, 3.4537, 4.8681   9.8754
U129 8 0, 0.2355, 0.8711, 1.5254, 2, 2.9050, 3.6799, 4.7831 9.8776
U130 8 0, 0.2907, 0.5858, 2, 2, 2.8061, 3.4142, 4.9032   9.8702
U131 8 0, 0.2679, 0.6571, 2, 2, 2.5293, 3.7321, 4.8136   9.8729
U132 8 0, 0.2243, 0.5858, 1.4108, 2.7273, 3, 3.4142, 4.6412 9.8113
U133 8 0, 0.3065, 0.3820, 1.6703, 2.6180, 3, 3.3297, 4.6935 9.8054
U134 8 0, 0.2137, 0.6177, 1.4977, 2.3537, 3, 3.8408, 4.4763 9.8138
U135 8 0, 0.1864, 1, 1, 2.4707, 3, 4, 4.3429   9.8196
U136 8 0, 0.1892, 0.8207, 1.2558, 2.2216, 3.3354, 3.7575, 4.4198   9.8191
U137 8 0, 0.2137, 0,6177, 1.4977, 2.3537, 3, 3.8408, 4.4763 9.8138
U138 8 0, 0.4915, 0.7530, 1.3204, 1.4450, 2.8258, 3.8019, 4.3623   10.001
U139 8 0, 0.3376, 1, 1.2426, 2.4249, 3, 3.4959, 4.4989   9.9758
U140 8 0, 0.2434, 1.1798, 1.3820, 2, 3.1386, 3.6180, 4.4383 9.9498
U141 8 0, 0.1930, 0.9231, 2, 2, 2.7890, 3.5143, 4.5806   9.9134
U142 8 0, 0.1667, 0.7276, 1.6353, 2.6729, 3, 3.5643, 4.2332 9.8524
U143 8 0, 0.5858, 0.5858, 2, 2, 3.4142, 3.4142, 4 10.0547

Table 3 : Partial proof of the conjecture.

On the other hand from (Tables 4-6), we can easily see that the energy E of U3=C4, U8=C5, U21=C6 and U54=C7 is maximal among all other unicyclic molecular graphs in the same number of vertices. But for n=8 the unicyclic molecular graph U143=C8 has energy 9.6568 which is less then energy of U140 and U142. Hence among the unicyclic molecular graphs the cycle Cn has maximal energy E for 3 ≤ n ≤ 7 but for n=8 it is not true.

# n Adjacency spectrum E
U1 3 2, 1, 1 4
U2 4 2.1701, 0.3111, 1:0000, 1:4812 4.9624
U3 4 2, 0, 0, 2 4
U4 5 2.3429, 0.4707, 0, 1:0000, 1:8136 5.6272
U5 5 2.3028, 0.6180, 0, 1:3028, 1:6180 5.8416
U6 5 2.2143, 1.0000, 0:5392, 1:0000, 1:6751 6.4286
U7 5 2.1358, 0.6622, 0, 0:6622, 2:1358 5.5959
U8 5 2, 0.6180, 0.6180, 1:6180, 1:6180 6.4721
U9 6 2.5141, 0.5720, 0, 0, 1, 2:0861 6.1723
U10 6 2.4458, 0.7968, 0, 0, 1:3703, 1:8723 6.4852
U11 6 2.4142, 0.6180, 0.6180, 0:4142, 1:6180, 1:6180 7.3006
U12 6 2.3799, 1, 0.2914, 0:7510, 1, -1.9202 7.3725
U13 6 2.2882, 0.8740, 0, 0, 0:8740, 2:2882 6.3246
U14 6 2.2784, 1.3174, 0, 0:7046, 1, 1:8912 7.1917
U15 6 2.3342, 1.0996, 0.2742, 5945, 1:3738, 1:7397 7.416
U16 6 2.2470, 0.8019, 0.5550, 0:5550, 0:8019, 2:2470 7.2078
U17 6 2.2361, 1, 0, 0, 1, 2:2361 6.4721
U18 6 2.2283, 1.3604, 0.1859, 1, 1, 1:7746 7.5492
U19 6 2.1753, 1.1260, 0, 0, 1:1260, 2:1753 6.6027
U20 6 2.1149, 1, 0.6180, 0:2541, 1:6180, 1:8608 7.4659
U21 6 2, 1, 1, 1, 1, 2 8
U22 7 2.6113, 0.6421, 0, 0, 0, 1, 2:3234 6.6468
U23 7 2.5944, 0.9159, 0, 0, 0, 1:3883, 2:1220 7.0206
U24 7 2.5616, 1, 0, 0, 0, 1:5616, 2 7.1231
U25 7 2.5374, 0.8493, 0.6180, 0, 04891, 1:6180, 1:8976 8.0095
U26 7 2.5450, 1, 0.4394, 0, 0:8302, 1, 2:1542 7.9688
U27 7 2.4495, 1, 0, 0, 0, 1, 2:4495 6.899
U28 7 2.4309, 1.3269, 0.3011, 0, 1, 1, 2:0590 8.1179
U29 7 2.6368, 1.5262, 0, 0, 0:7877, 1, 2:1071 7.7896
U30 7 2.4745, 1.1143, 0.5241, 0, 0:7615, 1:3891, 1:9624 8.2259
U31 7 2.4676, 1.1883, 0.3867, 0, 0:6043, 1:5274, 1:9108 8.0851
U32 7 2.3761, 1, 0.5952, 0, 0:5952, 1, 2:3761 7.9425
U33 7 2.3583, 1.1994, 0, 0, 0, 1:1994, 2:3583 7.1153
U34 7 2.4383, 1.1386, 0.6180, 0, 0:8202, 1:6180, 1:7566 8.3898
U35 7 2.3799, 1.4142, 0.2914, 0, 0:7510, 1:4142, 1:9202 8.1709
U36 7 2.3344, 1, 0.7420, 0, 0:7420, 1, 2:3344 8.1528
U37 7 2.3894, 1.3668, 0.3944, 0, 1, 1:1852, 1:9653 8.3011
U38 7 2.4142, 1, 1, 0:4142, 1, 1, 2 8.8284
U39 7 2.3244, 1.1472, 0.5304, 0, 0:5304, 1:1472, 2:3244 8.0038
U40 7 2.2562, 1.1899, 0.6180, 0, 0:3565, 1:6180, 2:0896 8.1283
U41 7 2.3623, 1.2470, 0.8258, 0:4450, 0:6796, 1:5085, 1:8019 8.8702
U42 7 2.3429, 1.4142, 0.4707, 0, 1, 1:4142, 1:8136 8.4556
U43 7 2.2970, 1.4933, 0.6400, 0:4631, 1, 1, 1:9672 8.8606
U44 7 2.2533, 1.6449, 0.2327, 0, 1, 1:2033, 1:9275 8.2616
U45 7 2.2764, 1.1859, 0.6416, 0, 0:6416, 1:1859, 2:2764 8.2078
U46 7 2.2638, 1.2793, 0.4883, 0, 0:4883, 1:2793, 2:2638 8.0629
U47 7 2.2361, 1.4142, 0, 0, 0, 1:4142, 2:2361 7.3006
U48 7 2.1987, 1.2470, 0.7135, 0, 0:4450, 1:8019, 1:9122 8.3184
    9  

Table 4: Partial proof of the conjecture.

# n Adjacency spectrum E
U49 7 2.2143, 1, 1, 0, 0:5392, 1:6751, 2 8.4286
U50 7 2.2332, 1.5643, 0.6729, 0:3647, 1, 1:2724, -1.8333 8.9408
U51 7 2.1889, 1.4142, 0.4569, 0, 0:4569, 1:4142, 2:1889 8.1199
U52 7 2.1515, 1.2685, 0.6180, 0.4206 , 0:8958, 1:6180, 1:9449 8.9174
U53 7 2.1010, 1.2593, 1, 0, 1, 1:2593, 2:1010 8.7206
U54 7 2, 1.2470, 1.2470, 0:4450, 0:4450, 1:8019, 1:8019 8.9879
U55 8 2.8434, 0.6932, 0, 0, 0, 0, 1, 2:5366 7.0732
U56 8 2.7448, 1, 0, 0, 0, 0, 1:3959, 2:3489 7.4896
U57 8 2.6872, 1.1408, 0, 0, 0, 0, 1:6396, 2:1885 7.6561
U58 8 2.6691, 1, 0.6180, 0, 0, 0:5240, 1:6180, 2:1451 8.5742
U59 8 2.6412, 1, 0.7237, 0, 0, 0:5892, 1:7757, 2 8.27298
U60 8 2.6131, 1.0824, 0, 0, 0, 0, 1:0824, 2:6131 7.391
U61 8 2.7073, 1, 0.5359, 0, 0, 0:8719, 1, 2:3713 8.4864
U62 8 2.4860, 1.6636, 0, 0, 0, 0:8360, 1, 2:3136 8.2992
U63 8 2.5178, 1.1380, 0.6045, 0, 0, 0:6045, 1:1380, 2:5178 8.5206
U64 8 2.4972, 1.3281, 0, 0, 0, 0, 1:3281, 2:4972 7.6506
U65 8 2.5860, 1.3350, 0.4559, 0, 0, 1, 1:1317, 2:2453 8.7539
U66 8 2.6095, 1.2598, 0.4468, 0, 0, 0:6084, 1:5710, 2:1367 8.6322
U67 8 2.6200, 1.1311, 0.6634, 0, 0, 0:8339, 1:3962, 2:1843 8.8289
U68 8 2.5009, 1.5558, 0.3044, 0, 0, 1, 1:1401, 2:2208 8.7223
U69 8 2.4812, 1.1701, 0.6889, 0, 0, 0:6889, 1:1701, 2:4812 8.6804
U70 8 2.4495, 1.4142, 0, 0, 0, 0, 1:4142, 2:4495 7.7274
U71 8 2.5831, 1.2346, 0.6180, 0, 0, 0:7631, 1:6180, 2:0545 8.8713
U72 8 2.5553, 1.1946, 0.7799, 0, 0, 0:8911, 1:7177, 1:9210 9.0596
U73 8 2.4495, 1, 1, 0, 0, 1, 1, 2:4495 8.899
U74 8 2.4412, 1.2124, 0.7555, 0, 0, 0:7555, 1:2124, 2:4412 88,182
U75 8 2.5141, 1.4142, 0.5720, 0, 0, 1, 1:4142, 2:0861 9.0006
U76 8 2.4465, 1.6383, 0.2976, 0, 0, 0:8262, 1:4347, 2:1216 8.7649
U77 8 2.4989, 1.4959, 0.4249, 0, 0, 0:7574, 1:6624, 2 8.8395
U78 8 2.5606, 1.1676, 0.6180, 0.5038, 0:3905, 0:8565, 1:6180, 1:9850 9.7
U79 8 2.4728, 1.4626, 0.6180, 0, 0, 1, 1:6180, 1:9354 9.1068
U80 8 2.4142, 1, 1, 0.4142, 0:4142, 1, 1, 2:4142 9.6568
U81 8 2.5714, 1, 1, 0.2754, 0:6379, 1, 1, 2:2116 9.699
U82 8 2.4142, 1.3028, 0.6180, 0, 0, 0:4142, 1:6180, 2:3028 8.67
U83 8 2.4812, 1.1701, 0.6889, 0, 0, 0:6889, 1:1701, 2:4812 8.6804
U84 8 2.5516, 1.3720, 0.5185, 0, 0, 1, 1:2658, 2:1762 8.8841
U85 8 2.4433, 1.5115, 0.6465, 0.2894, 0:5614, 1, 1:2249, 2:1044 9.7814
U86 8 2.3371, 1.2089, 1, 0, 0, 0:6699, 1:6975, 2:1786 9.092
U87 8 2.3113, 1.4269, 0.7353, 0, 0, 0:5338, 1:8365, 2:1032 8.947
U88 8 2.3761, 1.4142, 0.5952, 0, 0, 0:5952, 1:4142, 2:3761 8.771
U89 8 2.3268, 1.6080, 0, 0, 0, 0, 1:6080, 2:3268 7.8696
U90 8 2.4049, 1.2293, 0.6728, 0.5027, 0:5027, 0:6728, 1:2293, 2:4049 9.6194
U91 8 2.3867, 1.3497, 0.6941, 0, 0, 0:6941, 1:3497, 2:3867 8.861
U92 8 2.3968, 1.2665, 0.8069, 0, 0, 0:8069, 1:2665, 2:3968 8.9404
U93 8 2.3761, 1.4142, 0.5952, 0, 0, 0:5952, 1:4142, 2:4161 8.771
U94 8 2.4615, 0.3370, 1, 0, 0:4750, 1, 1:2118, 2:1118 9.5971
U95 8 2.4048, 1.6628, 0.4231, 0, 0, 1, 1:4446, 2:0461 8.9814
U96 8 2.4943, 1.2898, 0.8539, 0.2621, 0:5812, 0:7701, 1:5625, 1:9863 9.8002
    10  

Table 5 : Partial proof of the conjecture.

# n Adjacency spectrum E
U97 8 2.3914, 1.6180, 0.7729, 0, 0:6180, 1, 1, 2:1642 9.5645
U98 8 2.3028, 1.8608, 0.2541, 0, 0, 1, 1:3028, 2:1149 6.134722
U99 8 2.4812, 1.4142, 0.6889, 0, 0, 1:1701, 1:4142, 2 9.1686
U100 8 2.5019, 1.1643, 1, 0.2493, 0:4848, 1, 1:3965, 2:0341 9.8309
U101 8 2.4728, 1.4626, 0.6180, 0, 0, 1, 1:6180, 1:9354 9.1068
U102 8 2.4605, 1.2470, 1, 0.2391, 0:4450, 1, 1:6996, 1:8019 9.8931
U103 8 2.2904, 1.2470, 1, 0.3620, 0:4450, 0:5828, 1:8019, 2:0696 9.7987
U104 8 2.2784, 1.3174, 1, 0, 0, 0:7046, 1:8912, 2 9.1916
U105 8 2.3213, 1.4789, 0.6513, 0, 0, 0:6513, 1:4789, 2:3213 8.903
U106 8 2.3072, 1.5355, 0.5645, 0, 0, 0:5645, 1:5355, 2:3072 8.8144
U107 8 2.3583, 1.1993, 1, 0, 0, 1, 1:1993, 2:3583 9.1152
U108 8 2.3562, 1.2918, 0.7741, 0.4244, 0:4244, 0:7741, 1:2918, 2:3562 9.693
U109 8 2.3278, 1.6942, 0.7897, 0, 0:5017, 1, 1:2320, 2:0779 9.6233
U110 8 2.3920, 1.5739, 0.6852, 0.2715, 0:5010, 1, 1:4339, 1:9877 9.8452
U111 8 2.3577, 1.6931, 0.5273, 0, 0, 1:1628, 1:4708, -1.9445 9.1562
U112 8 2.4035, 1.4835, 0.9277, 0, 0:5022, 0:7790, 1:5934, 1:9401 9.6294
U113 8 2.4442, 1.4421, 0.6180, 0.4165, 0:3239, 1:1497, 1:6180, 1:8292 9.8416
U114 8 2.3920, 1.5739, 0.6852, 0.2715, 0:5010, 1, 1:4339, 1:9877 9.8452
U115 8 2.2361, 1.4142, 1, 0, 0, 1, 1:4142, 2:2361 9.3006
U116 8 2.2922, 1.3281, 0.6852, 0.6180, 0:2448, 0:9109, 1:6180, 2:1500 9.8472
U117 8 2.3344, 1.4142, 0.7420, 0, 0, 0:7420, 1:4142, 2:3344 8.9812
U118 8 2.3583, 1.1993, 1, 0, 0, 1, 1:1993, 2:3583 9.1152
U119 8 2.4227, 1.3726, 1, 0.1765, 0:6500, 1, 1:2897, 2:0321 9.9436
U120 8 2.1935, 1.2950, 1.1935, 0.2950, 0:2950, 1:1935, 1:2950, 2:1935 9.954
U121 8 2.1753, 1.4142, 1.1260, 0, 0, 1:1260, 1:4142, 2:1753 9.431
U122 8 2.1701, 1.4812, 1, 0.3111, 0:3111, 1, 1:4812, 2:1701 9.9248
U123 8 2.2105, 1.5047, 0.6180, 0.5043, 0, 1:1554, 1:6180, 2:0641 9.675
U124 8 2.2427, 1.2858, 1, 0.4410, 0:2266, 1, 1:6895, 2:0534 9.939
U125 8 2.2245, 1.4232, 0.8060, 0.5013, 0:2353, 0:9230, 1:8266, 1:9701 9.91
U126 8 2.2552, 1.5582, 0.6970, 0, 0, 0:6970, 1:5582, 2:2552 9.0208
U127 8 2.2143, 1.6751, 0.5392, 0, 0, 0:5392, 1:6751, 2:2143 8.8572
U128 8 2.2853, 1.4534, 0.6880, 0.4376, 0:4376, 0:6880, 0:4534, 2:2853 9.7286
U129 8 2.2725, 1.4924, 0.7801, 0, 0, 0:7801, 1:4924, 2:2725 9.09
U130 8 2.3028, 1.3028, 1, 0, 0, 1, 1:3028, 2:3028 9.2112
U131 8 2.2882, 1.4142, 0.8740, 0, 0, 0:8740, 1:4142, 2:2882 9.1528
U132 8 2.3028, 1.6180, 1, 0, 0:6180, 1, 1:3028, 2 9.8416
U133 8 2.3163, 1.5794, 1, 0.1346, 1, 1, 1, 2:0303 10.0606
U134 8 2.2623, 1.7571, 0.7790, 0.1832, 0:6696, 1, 1:3234, 1:9886 9.9632
U135 8 2.2429, 1.7928, 0.8048, 0, 0:4354, 1, 1:4573, 1:9478 9.681
U136 8 2.3455, 1.5988, 0.7784, 0.2449, 0:4197, 1:2246, 1:4638, 1:8596 9.9353
U137 8 2.3699, 1.4576, 1, 0.1724, 0:5771, 1, 1:5734, 1:8494 9.9998
U138 8 2.0912, 1.4427, 1.2470, 0.2163, 0:4450, 0:7764, 1:8019, 1:9738 9.9943
U139 8 2.1358, 1.4142, 1, 1.6621, 0:6621, 1, 1:4142, 2:1358 10.4242
U140 8 2.1648, 1.4739, 0.7691, 0.6180, 0:1627, 1:2635, 1:6180, 1:9815 10.0515
U141 8 2.1940, 1.5904, 0.8106, 0, 0, 0:8106, 1:5904, 2:1940 9.19
U142 8 2.2350, 1.6881, 1, 0.1326, 0:7386, 1, 1:4460, 1:8711 10.1114
U143 8 2, 1.4142, 1.4142, 0, 0, 1:4142, 1:4142, 2 9.6568

Table 6: Partial proof of the conjecture.

Conclusion

In this paper, our attention was focused on the Laplacian-energy-like invariant and energy of unicyclic molecular graphs with at most n vertices, where 3 ≤ n ≤ 8, and on the partial proof of the Conjecture 1.1. We have shown that Laplacian- energy-like invariant of cycle Cn is maximal among all other unicyclic molecular graphs for 3 ≤ n ≤ 8, a step towards the proof of the Conjecture 1.1.

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