Some stable and closed-shell structures of anticancer drugs by graph theoretical parameters

Abstract

The eigenvalues are significant in mathematics, but they are also relevant in other domains like as chemistry, economics, and a variety of others. In terms of our research, eigenvalues are used in chemistry to represent not only the form of energy but also the various physicochemical aspects of a chemical substance. We must comprehend the connection between mathematics and chemistry. The antibonding level is related to positive eigenvalues, the bonding level is associated to negative eigenvalues, and the nonbonding level is linked to zero eigenvalues. In this work, we studied some anticancer drug structures in terms of nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials. As a result, Carmustine, Caulibugulone-E, Aspidostomide-E anticancer drug structures are stable, closed-shell molecules since their nullity is equal to zero.

1. Introduction

The rapid proliferation of aberrant cells in the human body is known as cancer. Cancer-causing compounds are known as carcinogens. A carcinogen is a chemical compound found in cigarette smoke that contains certain components. It has the ability to spread throughout the body. A lump, abnormal bleeding, a prolonged cough, and weight loss are some of the symptoms of this condition. Chewing tobacco, obesity, a poor diet, laziness, and excessive alcohol consumption are the main causes of this cancerous condition. Several treatments, including surgery, radiotherapy, chemotherapy, hormone therapy, targeted therapy, and others, can be used to treat this severe disease. Anticancer medicines, which include alkylates and metabolites, are used to treat the disease known as cancer [1], [2], [3], [4].

Chemical graph theory is a branch of mathematics chemistry concerned with chemical graphs that depict chemical systems. The chemical graph theory allows for the definition of numerous anticancer medication properties [5], [6]. Several medicational structures are used in this study, by measuring nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials.

Nullity is an important factor in molecule stability, and if nullity is zero, the molecule is projected to have a stable, closed-shell electron configuration. The molecule is unstable, extremely reactive, nonexistent, and open shell if nullity is larger than zero [7]. As a result, some of the chosen anticancer drug structures are stable, closed-shell molecules since their nullity is equal to zero.

There are numerous research work is available on the nullity, matching number, eigenvalues of adjacency matrix, characteristics polynomials. Only recent and few important articles are given herewith their importance. A proof is given on the conjecture on the topic of nullity [8]. Rank four graphs are characterized in [9], rank five in [10]. For upper bounds on the nullity $n-2$ and $n-3$, see [11], while further generalizations are given in [12]. For a relation between matching number and rank of a graph given in [13]. Trees are discussed in terms of nullity of a graph found in [14]. On the nullity of a graph with cut-points [15]. Unicyclic graphs in terms of nullity and matching number are found in [16], while for bicyclci graphs are herein [17] and for the line operation of unicyclic graphs and their nullity in [18]. Pure mathematical and abstract theory on nullity is available in [19], [20], [21]. Nullity is expressed in terms of maximum degree of a vertex [22], [23]. Authors of [24], [25], computed the double metric resolvability of convex polytopes, authors of [26], computed the edge version of resolvability and double resolvability of some generalized graphs.

The chemical complex is typically depicted as a graph, with the elements representing vertices and the bonds linking them representing edges. Similarly, the anticancer medications under investigation are treated as chemical compounds, and the parameters are investigated [27]. Graph theory provides methods such as QSAR, QSPR, and QSTR (quantitative structure-activity/property/toxicity relationship) that chemists and pharmacists can employ to enhance their research. Drugs are represented as molecular networks in theoretical chemistry, with each vertex representing an atom and each edge representing a relationship between two atoms [28], [29]. Assume that $G(V,E)$ is a molecular graph with vertex and edge sets. Simple graphs with no cycle creation and several edges are considered, while $|V\left(G\right)|=n$ and $|E\left(G\right)|=m$ are order and size of a graph G, respectively. Adjacency matrix is defined by the elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph and it is denoted by $A\left(G\right)$. Characteristics polynomial of a graph is defined by $\mathrm{\text{Char}}(A\left(G\right);\lambda )=\mathrm{\text{det}}(A\left(G\right)-\lambda I)=0$, where I is an identity matrix of order same as A-matrix. The values of parameter λ in the equation $\mathrm{\text{det}}(A\left(G\right)-\lambda I)=0$ are known as eigenvalues and we symbolized as $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G\right);\lambda )\right)$. While the mulitplicity of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G\right);\lambda )\right)=0$ is known as nullity of a graph G, and usually it is denoted by $\eta \left(G\right)$ [30]. In [31], the nullity of a bipartite graph is defined by

$$\eta \left(G\right)=n-2M\left(G\right).$$ (1)

In the Equation (1), a parameter $M\left(G\right)$ is known as matching number and defined by the size of a largest maximal independent edge set [32], [33]. In Fig. 1 to Fig. 10, the wavy edges denoted that edges contributed towards the count of matching number for all anticancer drug structures.

Figure 1.

Anticancer drug structure Amathas-Piramide-E.

Figure 10.

Anticancer drug structure Tambjamine-K.

The positive inertia index is represented by the number of positive eigenvalues $p(G)$, whereas the negative inertia index is represented by the number of negative eigenvalues $q(G)$. An energy of the graph G is the absolute sum of all the eigenvalues. It is written as $E={\sum }_{i=1}^k=|{\lambda }_i|$ in mathematics [7], [34].

2. Results on matching number and nullity of anticancer drug structures

Several medicational structures are used in this study, by measuring nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials.

2.1. Results of matching number and nullity of Amathas-Piramide-E anticancer drug structure

The order of graph obtained from Amathas-Piramide-E $|V\left(G_1\right)|=22$, while the size is counted in $|E\left(G_1\right)|=24$. Moreover, the vertex and edge set is defined by

$$V(G_1)=\{v_i:1\le i\le 22\},E(G_1)=\{v_i{v}_j:i,j=1,2,\dots ,7\}\cup \{v_i{v}_j:i,j=8,9,\dots ,15\}\cup \{v_{14}{v}_{15},v_{11}{v}_{16},v_8{v}_{16},v_6{v}_{12},v_5{v}_{19},v_{12}{v}_{20},v_{14}{v}_{21},v_{15}{v}_{22},v_1{v}_{17},v_3{v}_{18}\}$$

Lemma 2.1

Let $G_1$ be a graph obtained by an anticancer drug structure Amathas-Piramide-E. Then $\eta \left(G_1\right)=1$ .

Proof

Observe that $G_1$ is the graph of Amathas-Piramide-E an anticancer drug structure developed by some pendant vertices, two pentagons, and a single hexagon. Fig. 1, shows that it is not a bipartite graph. So to compute the nullity of $G_1$, we can not follow the Equation (1). Therefore, using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_1\right);\lambda )$ of the adjacency matrix $A\left(G_1\right)$ of graph $G_1$, characteristics polynomial is described as;

$$\begin{gathered} \mathrm{\text{Char}}(A\left(G_1\right);\lambda )={\lambda }^{22}-23{\lambda }^{20}+220{\lambda }^{18}-2{\lambda }^{17}-1146{\lambda }^{16}+32{\lambda }^{15}+3577{\lambda }^{14}-200{\lambda }^{13}-6932{\lambda }^{12} \\ +632{\lambda }^{11}+8353{\lambda }^{10}-1092{\lambda }^9-6088{\lambda }^8+1040{\lambda }^7+2513{\lambda }^6-518{\lambda }^5-507{\lambda }^4 \\ +114{\lambda }^3+33{\lambda }^2-6\lambda . \end{gathered}$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_1\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_1\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_1\right);\lambda )\right)=\{0,-2.4727,-2.1834,-1.7631,-1.6180,-1.6063,-1.3109,-0.9348,-0.8836,-0.7166,-0.3071,0.1585,0.5259,0.6180,0.7480,0.9260,1.2615,1.3524,1.6064,1.8599,2.2338,2.5061\}.$$

Observing that there is only a single value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_1\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_1\right)=1$.

Furthermore, the Fig. 1, it is showing that there are ten counts of wavy edges and which is the matching number $M\left(G_1\right)$ of graph $G_1$. □

2.2. Results of matching number and nullity of Carmustine anticancer drug structure

The order of graph obtained from Carmustine $|V\left(G_2\right)|=12$, while the size is counted in $|E\left(G_2\right)|=11$. Moreover, the vertex and edge set is defined by

$$V(G_2)=\{v_i:1\le i\le 12\},E(G_2)=\{v_i{v}_j:i,j=1,2,\dots ,9\}\cup \{v_5{v}_{12},v_4{v}_{10},v_{10}{v}_{11}\}$$

Lemma 2.2

Let$G_2$be a graph obtained by an anticancer drug structure Carmustine. Then$\eta \left(G_2\right)=0$.

Proof

Observe that $G_2$ is the graph of Carmustine an anticancer drug structure developed by some pendant vertices, without any cycle. Fig. 2, shows that it is a bipartite graph. So to compute the nullity of $G_2$, we will follow the Equation (1). Fig. 1, it is showing that there is six-count of wavy edges and which is the matching number $M\left(G_2\right)$ of graph $G_2$. So by applying the definition of the nullity of a bipartite graph $\eta \left(G_2\right)=n-2M=12-2(6)=0$.

Furthermore, by using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_2\right);\lambda )$ of the adjacency matrix $A\left(G_2\right)$ of graph $G_2$, characteristics polynomial is described as;

$$\mathrm{\text{Char}}(A\left(G_2\right);\lambda )={\lambda }^{12}-11{\lambda }^{10}+43{\lambda }^8-74{\lambda }^6+55{\lambda }^4-14{\lambda }^2+1.$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_2\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_2\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_1\right);\lambda )\right)=\{-2.1673,-1.6783,-1.3427,-1.1358,-0.5246,-0.3436,0.3436,0.5246,1.1358,1.3427,1.6783,2.1673\}.$$

Observing that there is no value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_2\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_2\right)=0$. □

Figure 2.

Anticancer drug structure Carmustine.

2.3. Results of matching number and nullity of Caulibugulone-E anticancer drug structure

The order of graph obtained from Caulibugulone-E $|V\left(G_3\right)|=14$, while the size is counted in $|E\left(G_3\right)|=15$. Moreover, the vertex and edge set is defined by

$$V(G_3)=\{v_i:1\le i\le 14\},E(G_3)=\{v_i{v}_j:i,j=1,2,\dots ,12\}\cup \{v_4{v}_{14},v_3{v}_{12},v_5{v}_{10},v_{11}{v}_{13}\}$$

Lemma 2.3

Let $G_3$ be a graph obtained by an anticancer drug structure Caulibugulone-E. Then $\eta \left(G_3\right)=0$ .

Proof

Observe that $G_3$ is the graph of Caulibugulone-E an anticancer drug structure developed by some pendant vertices, two cycles of length eight. Fig. 3, shows that it is a bipartite graph. So to compute the nullity of $G_3$, we will follow the Equation (1). Fig. 3, it is showing that there is seven-count of wavy edges and which is the matching number $M\left(G_3\right)$ of graph $G_3$. So by applying the definition of the nullity of a bipartite graph $\eta \left(G_3\right)=n-2M=14-2(7)=0$.

Furthermore, by using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_3\right);\lambda )$ of the adjacency matrix $A\left(G_3\right)$ of graph $G_3$, characteristics polynomial is described as;

$$\mathrm{\text{Char}}(A\left(G_3\right);\lambda )={\lambda }^{14}-15{\lambda }^{12}+84{\lambda }^{10}-225{\lambda }^8+304{\lambda }^6-200{\lambda }^4+56{\lambda }^2-4.$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_3\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_3\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_3\right);\lambda )\right)=\{-2.4344,-1.8478,-1.6055,-1.2736,-0.8769,-0.7654,-0.3240,0.3240,0.7654,0.8769,1.2736,1.6055,1.8478,2.4344\}.$$

Observing that there is no value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_3\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_3\right)=0$. □

Figure 3.

Anticancer drug structure Caulibugulone-E.

2.4. Results of matching number and nullity of Aspidostomide-E anticancer drug structure

The order of graph obtained from Aspidostomide-E $|V\left(G_4\right)|=26$, while the size is counted in $|E\left(G_4\right)|=29$. Moreover, the vertex and edge set is defined by

$$V(G_4)=\{v_i:1\le i\le 26\},E(G_4)=\{v_i{v}_j:i,j=1,2,\dots ,11\}\cup \{v_i{v}_j:i,j=12,13,\dots ,20\}\cup \{v_3{v}_{11},v_2{v}_3,v_1{v}_2,v_5{v}_{22},v_6{v}_{10},v_7{v}_{23},v_8{v}_{24},v_9{v}_{25},v_{11}{v}_{15},v_{12}{v}_{21},v_{12}{v}_{20},v_{14}{v}_{18},v_{16}{v}_{26}\}$$

Lemma 2.4

Let $G_4$ be a graph obtained by an anticancer drug structure Aspidostomide-E. Then $\eta \left(G_4\right)=0$ .

Proof

Observe that $G_4$ is the graph of Aspidostomide-E an anticancer drug structure developed by some pendant vertices, two pentagons the same count of hexagons. Fig. 4, shows that it is not a bipartite graph. So to compute the nullity of $G_4$, we can not follow the Equation (1). Therefore, using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_4\right);\lambda )$ of the adjacency matrix $A\left(G_4\right)$ of graph $G_4$, characteristics polynomial is described as;

$$\begin{gathered} \mathrm{\text{Char}}(A\left(G_4\right);\lambda )={\lambda }^{26}-29{\lambda }^{24}+361{\lambda }^{22}-4{\lambda }^{21}-2536{\lambda }^{20}+78{\lambda }^{19}+11121{\lambda }^{18}-618{\lambda }^{17}-31803{\lambda }^{16} \\ +2588{\lambda }^{15}+60177{\lambda }^{14}-6230{\lambda }^{13}-74914{\lambda }^{12}+8780{\lambda }^{11}+59908{\lambda }^{10}-7076{\lambda }^9 \\ -29435{\lambda }^8+3058{\lambda }^7+8260{\lambda }^6-628{\lambda }^5-1177{\lambda }^4+50{\lambda }^3+69{\lambda }^2-1. \end{gathered}$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_4\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_4\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_4\right);\lambda )\right)=\{-2.4787,-2.2370,-2.1237,-1.9502,-1.7048,-1.5638,-1.3400,-1.0401,-0.8788,-0.5584,-0.5177,-0.2512,-0.1650,0.1340,0.3667,0.4500,0.7540,1.0000,1.1111,1.2313,1.3402,1.6019,1.8261,1.9975,2.3929,2.6038\}.$$

Observing that there is no value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_4\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_4\right)=0$.

Furthermore, the Fig. 4, it is showing that there are 13 counts of wavy edges and which is the matching number $M\left(G_4\right)$ of graph $G_4$. □

Figure 4.

Anticancer drug structure Aspidostomide-E.

2.5. Results of matching number and nullity of Convolutamide-A anticancer drug structure

The order of graph obtained from Convolutamide-A $|V\left(G_5\right)|=31$, while the size is counted in $|E\left(G_5\right)|=32$. Moreover, the vertex and edge set is defined by

$$V(G_5)=\{v_i:1\le i\le 31\},E(G_5)=\{v_i{v}_j:i,j=1,2,\dots ,6\}\cup \{v_i{v}_j:i,j=7,8,\dots ,11\}\cup \{v_i{v}_j:i,j=12,13,\dots ,25\}\cup \{v_1{v}_{30},v_1{v}_6,v_6{v}_{29},v_5{v}_{28},v_3{v}_{11},v_{11}{v}_{27},v_{10}{v}_{26},v_7{v}_{11},v_9{v}_{12},v_{12}{v}_{31}\}$$

Lemma 2.5

Let $G_5$ be a graph obtained by an anticancer drug structure Convolutamide-A. Then $\eta \left(G_5\right)=3$ .

Proof

Observe that $G_5$ is the graph of Convolutamide-A an anticancer drug structure developed by some pendant vertices, 1 pentagon, and a single hexagon. Fig. 5, shows that it is not a bipartite graph. So to compute the nullity of $G_5$, we can not follow the Equation (1). Therefore, using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_5\right);\lambda )$ of the adjacency matrix $A\left(G_5\right)$ of graph $G_5$, characteristics polynomial is described as;

$$\begin{gathered} \mathrm{\text{Char}}(A\left(G_5\right);\lambda )={\lambda }^{31}-32{\lambda }^{29}+453{\lambda }^{27}-2{\lambda }^{26}-3749{\lambda }^{25}+46{\lambda }^{24}+20191{\lambda }^{23}-456{\lambda }^{22}-74531{\lambda }^{21} \\ +2560{\lambda }^{20}+193407{\lambda }^{19}-8988{\lambda }^{18}-356043{\lambda }^{17}+20548{\lambda }^{16}+463363{\lambda }^{15}-30848{\lambda }^{14} \\ -419835{\lambda }^{13}+29904{\lambda }^{12}+257357{\lambda }^{11}-17904{\lambda }^{10}-101942{\lambda }^9+6096{\lambda }^8+24282{\lambda }^7 \\ -1024{\lambda }^6-3074{\lambda }^5+64{\lambda }^4+156{\lambda }^3. \end{gathered}$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_5\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_5\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_5\right);\lambda )\right)=\{0,0,0,-2.4356,-2.1859,-2.0217,-1.9405,-1.7932,-1.5856,-1.4142,-1.3550,-1.2082,-1.0995,-0.7777,-0.5774,-0.4289,-0.3963,0.3925,0.4613,0.7137,0.8663,1.0000,1.1379,1.3320,1.4142,1.5357,1.7466,1.8915,1.9701,2.2327,2.5248\}.$$

Observing that there are three count of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_1\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_5\right)=3$.

Furthermore, the Fig. 5, it is showing that there are fourteen counts of wavy edges and which is the matching number $M\left(G_5\right)$ of graph $G_5$. □

Figure 5.

Anticancer drug structure Convolutamide-A.

2.6. Results of matching number and nullity of Convolutamine-F anticancer drug structure

The order of graph obtained from Convolutamine-F $|V\left(G_6\right)|=15$, while the size is counted in $|E\left(G_6\right)|=15$. Moreover, the vertex and edge set is defined by

$$V(G_6)=\{v_i:1\le i\le 15\},E(G_6)=\{v_i{v}_j:i,j=1,2,\dots ,6\}\cup \{v_i{v}_j:i,j=7,8,\dots ,10\}\cup \{v_1{v}_{12},v_2{v}_{14},v_{13}{v}_{14},v_3{v}_{15},v_4{v}_7,v_5{v}_{11},v_1{v}_6,v_3{v}_4\}$$

Lemma 2.6

Let $G_6$ be a graph obtained by an anticancer drug structure Convolutamine-F. Then $\eta \left(G_6\right)=3$ .

Proof

Observe that $G_3$ is the graph of Convolutamine-F an anticancer drug structure developed by some pendant vertices, single cycle of length eight. Fig. 6, shows that it is a bipartite graph. So to compute the nullity of $G_6$, we will follow the Equation (1). Fig. 6, it is showing that there is six-count of wavy edges and which is the matching number $M\left(G_6\right)$ of graph $G_6$. So by applying the definition of the nullity of a bipartite graph $\eta \left(G_6\right)=n-2M=15-2(6)=3$.

Furthermore, by using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_6\right);\lambda )$ of the adjacency matrix $A\left(G_6\right)$ of graph $G_6$, characteristics polynomial is described as;

$$\mathrm{\text{Char}}(A\left(G_6\right);\lambda )={\lambda }^{15}-15{\lambda }^{13}+85{\lambda }^{11}-233{\lambda }^9+323{\lambda }^7-211{\lambda }^5+50{\lambda }^3.$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_6\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_6\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_6\right);\lambda )\right)=\{0,0,0,-2.3991,-1.8061,-1.5606,-1.4142,-1.0000,-0.7394,0.7394,1.0000,1.4142,1.5606,1.8061,2.3991\}.$$

Observing that there are three values of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_2\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_6\right)=3$. □

Figure 6.

Anticancer drug structure Convolutamine-F.

2.7. Results of matching number and nullity of Convolutamydine-A anticancer drug structure

The order of graph obtained from Convolutamydine-A $|V\left(G_7\right)|=16$, while the size is counted in $|E\left(G_7\right)|=18$. Moreover, the vertex and edge set is defined by

$$V(G_7)=\{v_i:1\le i\le 16\},E(G_7)=\{v_i{v}_j:i,j=1,2,\dots ,12\}\cup \{v_2{v}_{14},v_4{v}_{15},v_6{v}_{16},v_{12}{v}_{13},v_3{v}_{10},v_4{v}_8,v_1{v}_{12}\}$$

Lemma 2.7

Let $G_7$ be a graph obtained by an anticancer drug structure Convolutamydine-A. Then $\eta \left(G_7\right)=4$ .

Proof

Observe that $G_7$ is the graph of Convolutamydine-A an anticancer drug structure developed by some pendant vertices, two pentagons, and a single hexagon. Fig. 7, shows that it is not a bipartite graph. So to compute the nullity of $G_7$, we can not follow the Equation (1). Therefore, using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_7\right);\lambda )$ of the adjacency matrix $A\left(G_7\right)$ of graph $G_7$, characteristics polynomial is described as;

$$\mathrm{\text{Char}}(A\left(G_7\right);\lambda )={\lambda }^{16}-18{\lambda }^{14}+124{\lambda }^{12}-4{\lambda }^{11}-415{\lambda }^{10}+34{\lambda }^9+700{\lambda }^8-90{\lambda }^7-542{\lambda }^6+76{\lambda }^5+133{\lambda }^4.$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_7\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_7\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_7\right);\lambda )\right)=\{0,0,0,0,-2.3958,-2.1148,-1.8930,-1.4927,-1.3535,-0.5272,0.9296,1.1075,1.4992,1.5584,2.0790,2.6033\}.$$

Observing that there are four counts of the values of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_1\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_7\right)=4$.

Furthermore, the Fig. 7, it is showing that there are six counts of wavy edges and which is the matching number $M\left(G_7\right)$ of graph $G_7$. □

Figure 7.

Anticancer drug structure Convolutamydine-A.

2.8. Results of matching number and nullity of Perfragilin-A anticancer drug structure

The order of graph obtained from Perfragilin-A $|V\left(G_8\right)|=17$, while the size is counted in $|E\left(G_8\right)|=18$. Moreover, the vertex and edge set is defined by

$$V(G_8)=\{v_i:1\le i\le 17\},E(G_8)=\{v_i{v}_j:i,j=1,2,\dots ,13\}\cup \{v_1{v}_{11},v_{11}{v}_{12},v_5{v}_6,v_2{v}_{13},v_5{v}_{14},v_6{v}_{15},v_9{v}_{16},v_{10}{v}_{17},v_1{v}_{10},v_3{v}_8\}$$

Lemma 2.8

Let $G_8$ be a graph obtained by an anticancer drug structure Perfragilin-A. Then $\eta \left(G_8\right)=1$ .

Proof

Observe that $G_8$ is the graph of Perfragilin-A an anticancer drug structure developed by some pendant vertices, two cycles of length eight. Fig. 8, shows that it is a bipartite graph. So to compute the nullity of $G_8$, we will follow the Equation (1). Fig. 8, it is showing that there is eight-count of wavy edges and which is the matching number $M\left(G_8\right)$ of graph $G_8$. So by applying the definition of the nullity of a bipartite graph $\eta \left(G_8\right)=n-2M=17-2(8)=1$.

Furthermore, by using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_8\right);\lambda )$ of the adjacency matrix $A\left(G_8\right)$ of graph $G_8$, characteristics polynomial is described as;

$$\mathrm{\text{Char}}(A\left(G_8\right);\lambda )={\lambda }^{17}-18{\lambda }^{15}+126{\lambda }^{13}-442{\lambda }^{11}+833{\lambda }^9-836{\lambda }^7+413{\lambda }^5-82{\lambda }^3+4\lambda .$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_8\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_8\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_8\right);\lambda )\right)=\{0,-2.4978,-2.0576,-1.6713,-1.4839,-1.1368,-0.9232,-0.5633,-0.2654,0.2654,0.5633,0.9232,1.1368,1.4839,1.6713,2.0576,2.4978\}.$$

Observing that there is a single value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_8\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_8\right)=1$. □

Figure 8.

Anticancer drug structure Perfragilin-A.

2.9. Results of matching number and nullity of Melatonin anticancer drug structure

The order of graph obtained from Melatonin $|V\left(G_9\right)|=17$, while the size is counted in $|E\left(G_9\right)|=18$. Moreover, the vertex and edge set is defined by

$$V(G_9)=\{v_i:1\le i\le 17\},E(G_9)=\{v_i{v}_j:i,j=1,2,\dots ,9\}\cup \{v_i{v}_j:i,j=10,11,\dots ,14\}\cup \{v_{13}{v}_{15},v_4{v}_8,v_2{v}_{16},v_{16}{v}_{17}\}$$

Lemma 2.9

Let $G_9$ be a graph obtained by an anticancer drug structure Melatonin. Then $\eta \left(G_9\right)=1$ .

Proof

Observe that $G_9$ is the graph of Melatonin an anticancer drug structure developed by some pendant vertices, a single pentagon, and a single hexagon. Fig. 9, shows that it is not a bipartite graph. So to compute the nullity of $G_9$, we can not follow the Equation (1). Therefore, using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_9\right);\lambda )$ of the adjacency matrix $A\left(G_9\right)$ of graph $G_1$, characteristics polynomial is described as;

$$\begin{gathered} \mathrm{\text{Char}}(A\left(G_9\right);\lambda )={\lambda }^{17}-18{\lambda }^{15}+129{\lambda }^{13}-2{\lambda }^{12}-474{\lambda }^{11}+20{\lambda }^{10}+957{\lambda }^9-72{\lambda }^8-1057{\lambda }^7+114{\lambda }^6 \\ +598{\lambda }^5-80{\lambda }^4-144{\lambda }^3+20{\lambda }^2+8\lambda . \end{gathered}$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_9\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_9\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_9\right);\lambda )\right)=\{0,-2.2999,-2.0000,-1.7964,-1.6653,-1.1300,-1.0000,-0.7151,-0.1949,0.4233,0.6396,1.0000,1.0888,1.4530,1.8199,1.9558,2.4211\}.$$

Observing that there is only a single value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_9\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_9\right)=1$.

Furthermore, the Fig. 9, it is showing that there are eight counts of wavy edges and which is the matching number $M\left(G_9\right)$ of graph $G_9$. □

Figure 9.

Anticancer drug structure Melatonin.

2.10. Results of matching number and nullity of Tambjamine-K anticancer drug structure

The order of graph obtained from Tambjamine-K $|V\left(G_{10}\right)|=19$, while the size is counted in $|E\left(G_{10}\right)|=20$. Moreover, the vertex and edge set is defined by

$$V(G_{10})=\{v_i:1\le i\le 19\},E(G_{10})=\{v_i{v}_j:i,j=1,2,\dots ,5\}\cup \{v_i{v}_j:i,j=6,7,\dots ,10\}\cup \{v_i{v}_j:i,j=11,12,\dots ,16\}\cup \{v_3{v}_6,v_8{v}_{18},v_{18}{v}_{19},v_9{v}_{11},v_{15}{v}_{17}\}$$

Lemma 2.10

Let $G_{10}$ be a graph obtained by an anticancer drug structure Tambjamine-K. Then $\eta \left(G_{10}\right)=1$ .

Proof

Observe that $G_{10}$ is the graph of Tambjamine-K an anticancer drug structure developed by some pendant vertices, and two pentagons. Fig. 10, shows that it is not a bipartite graph. So to compute the nullity of $G_{10}$, we can not follow the Equation (1). Therefore, using the definition of nullity, we will determine the characteristics polynomial $\mathrm{\text{Char}}(A\left(G_{10}\right);\lambda )$ of the adjacency matrix $A\left(G_{10}\right)$ of graph $G_{10}$, characteristics polynomial is described as;

$$\begin{gathered} \mathrm{\text{Char}}(A\left(G_{10}\right);\lambda )={\lambda }^{19}-20{\lambda }^{17}+164{\lambda }^{15}-4{\lambda }^{14}-714{\lambda }^{13}+52{\lambda }^{12}+1785{\lambda }^{11}-258{\lambda }^{10}-2581{\lambda }^9+612{\lambda }^8 \\ +2045{\lambda }^7-710{\lambda }^6-755{\lambda }^5+360{\lambda }^4+71{\lambda }^3-54{\lambda }^2+6\lambda . \end{gathered}$$

Now, by solving $\mathrm{\text{Char}}(A\left(G_{10}\right);\lambda )=0$ for λ, the eigenvalues $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_{10}\right);\lambda )\right)$ of determined polynomials are

$$\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_{10}\right);\lambda )\right)=\{0,-2.1363,-2.1225,-1.9049,-1.6180,-1.5051,-1.2730,-0.9461,-0.4990,0.1673,0.4002,0.6012,0.6180,1.0000,1.3453,1.4991,1.9169,2.0555,2.4014\}.$$

Observing that there is only a single value of $\mathrm{\text{Eig}}\left(\mathrm{\text{Char}}(A\left(G_{10}\right);\lambda )\right)=0$ and which is concluding that $\eta \left(G_{10}\right)=1$.

Furthermore, the Fig. 10, it is showing that there are eight counts of wavy edges and which is the matching number $M\left(G_{10}\right)$ of graph $G_{10}$. □

3. Conclusion and discussion

Some anticancer drug structures are studied namely, Amathas-Piramide-E, Carmustine, Caulibugulone-E, Aspidostomide-E, Convolutamide-A, Convolutamine-F, Convolutamydine-A, Perfragilin-A, Melatonin, and Tambjamine-K. All these structures are studied in terms of nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials. As a result, Carmustine, Caulibugulone-E, Aspidostomide-E anticancer drug structures are stable, closed-shell molecules since their nullity is equal to zero (Table 1).

Table 1.

Energy, positive-negative-Inertia and nullity for various anticancer drug structures.

G	η(G)	p(G)	q(G)	E
G1	1	11	10	27.5929
G2	0	12	6	14.3847
G3	0	7	7	18.2550
G4	0	13	13	33.6189
G5	3	14	14	38.4390
G6	3	6	6	17.8389
G7	4	6	6	19.5540
G8	1	8	8	21.1985
G9	1	8	8	21.6032
G10	1	10	8	24.0097

The first Betti number $b_1(G)=m+|C|-n$. It is also called the cyclomatic number a term introduced by [35]. While in [36], nullity termed referred as first Betti number or cyclomatic number ($cn(G)$). Table 2, compare nullity, first Betti number and cyclomatic number for anticancer drug structures.

Table 2.

Different parameters for various anticancer drug structures.

G	η(G)	b1(G)	cn(G)
G1	1	3	3
G2	0	0	0
G3	0	2	2
G4	0	4	4
G5	3	2	2
G6	3	1	1
G7	4	3	3
G8	1	2	2
G9	1	2	2
G10	1	2	2

From the comparison given in the Table 2, the first betti number and nullity is equal for $G_2$, otherwise all the anticancer drug structures have different nullity and first Betti number, somehow the count of cyclomatic number and first Betti number are same for all the structures. Moreover, further research direction can be considered for the structures of [37], in the context to this chosen topic.

CRediT authorship contribution statement

Ali. N. A. Koam, A. Ahmad, M. Azeem, Khalil Hadi Hakami, and Kashif Elahi: conceived and designed the experiments; performed the experiments; analyzed and interpreted the data; contributed reagents, materials, analysis tools or data; wrote the paper.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data included in article/supplementary material/referenced in article.