Predictive modeling for physicochemical properties of -lactam antibiotics through eigenvalue based topological indices and non linear regression techniques

Abstract

In this work, a quantitative structure property relationship analysis is performed for selected -lactam antibiotics using eigenvalue based spectral descriptors derived from molecular graph representations. Molecular structures are modeled as graphs, and spectral indices obtained from adjacency, Laplacian, signless Laplacian, and distance matrices are employed to encode global topological characteristics. Non linear regression models, including quadratic, logarithmic, and power forms, are used to investigate relationships between spectral descriptors and physicochemical properties such as boiling point, molar volume, molar refractivity, polar surface area, and surface tension. The results indicate that several spectral descriptors exhibit meaningful correlations with properties primarily governed by molecular size and connectivity. Among the considered models, quadratic regression generally shows marginally improved performance, reflected by higher coefficients of determination and lower prediction errors. In contrast, properties strongly influenced by electronic and surface specific effects display weaker correlations, highlighting the intrinsic limitations of purely graph based descriptors. Overall, the findings demonstrate the applicability of spectral graph descriptors as interpretable tools for QSPR modeling of -lactam antibiotics and provide a basis for further studies involving larger datasets and complementary descriptor classes.

Introduction

-lactam antibiotics constitute one of the most widely used and clinically important classes of antibacterial agents. Their antibacterial activity arises from the presence of the four membered -lactam ring, which binds to penicillin binding proteins (PBPs) and inhibits the synthesis of peptidoglycan in bacterial cell walls, ultimately leading to bacterial cell lysis. Owing to their broad spectrum efficacy, structural versatility, and comparatively low toxicity, -lactam antibiotics continue to play a central role in the treatment of bacterial infections¹⁸.

The -lactam family comprises several therapeutically significant subclasses, including penicillins, cephalosporins, carbapenems, and -lactamase inhibitors. Despite their long standing clinical success, extensive and prolonged use of -lactam antibiotics has contributed to the emergence of antibiotic resistant bacterial strains. This growing resistance challenge highlights the need for systematic approaches to understand and predict the physicochemical behavior of -lactam compounds. In this context, computational modeling techniques offer valuable complementary tools for analyzing molecular properties and supporting drug development and optimization efforts¹¹.

Quantitative structure property relationship (QSPR) modeling provides a well-established computational framework for correlating molecular structure with experimentally measurable physicochemical properties using numerical descriptors. Within chemical graph theory, molecular structures are represented as graphs in which atoms correspond to vertices and chemical bonds correspond to edges. This representation enables the construction of topological descriptors that encode essential structural features such as molecular size, connectivity, branching, and cyclicity. Such descriptors have been extensively applied in QSPR studies to model physicochemical, pharmacokinetic, and biological properties of chemical compounds^11,29,32.

From a chemical perspective, graph associated matrices provide complementary representations of molecular structure at different levels of connectivity and organization. The adjacency matrix captures direct bonding relationships between atoms and reflects local connectivity patterns within a molecule. The Laplacian and signless Laplacian matrices incorporate both adjacency information and vertex degrees, thereby encoding the interplay between local bonding environments and global molecular structure. In contrast, the distance matrix records shortest path distances between all pairs of atoms and reflects global molecular size and shape. Spectral descriptors derived from these matrices are therefore sensitive to variations in molecular topology and structural organization^{12,21,22,26,27}.

Eigenvalue based spectral descriptors obtained from graph associated matrices have received increasing attention as alternatives to traditional degree based or distance based topological indices. By incorporating information from the entire molecular graph, spectral descriptors capture both local and global structural characteristics and are particularly responsive to subtle changes in molecular architecture. This enhanced sensitivity makes spectral indices especially suitable for modeling structurally complex drug molecules, where small structural modifications can lead to pronounced changes in physicochemical properties^19,20.

In the present study, seven -lactam antibiotics are selected to represent structurally diverse and clinically relevant members of this drug class, namely Amoxicillin, Oxacillin, Cefuroxime, Meropenem, Imipenem, Ertapenem, and Clavulanic acid. These compounds differ in molecular size, side chain complexity, and substitution patterns, allowing an assessment of how eigenvalue based spectral descriptors respond to structural diversity within a chemically coherent group. Reliable experimental physicochemical property data are available for all selected antibiotics, ensuring consistency and comparability in QSPR modeling (Fig. 1)¹⁸.

Fig. 1.

Molecular Structure of -lactam antibiotics.

Motivated by these considerations, this work focuses on the systematic application of eigenvalue based spectral descriptors derived from adjacency, Laplacian, signless Laplacian, and distance matrices for QSPR analysis of -lactam antibiotics. Interpretable non linear regression models are employed to investigate relationships between molecular topology and key physicochemical properties. By integrating spectral graph theory with QSPR modeling, the study provides insight into the capability of eigenvalue based descriptors to capture structure property relationships in -lactam antibiotics.

Eigenvalue based topological indices

In quantitative structure property relationship (QSPR) modeling, molecular structures are commonly represented using graph theoretic concepts, which enable structural information to be encoded into mathematically well defined numerical descriptors. In this section, we describe the construction of eigenvalue based spectral topological indices employed in the present study.

Let be a simple and connected molecular graph associated with a chemical compound, where the vertex set V(G) represents atoms and the edge set E(G) represents covalent bonds. Let and denote the number of vertices and edges of G, respectively¹⁰. This graph representation forms the basis for the spectral descriptors used in the QSPR analysis.

The structural characteristics of the molecular graph G are captured using graph-associated matrices. The adjacency matrix is defined by

Let D(G) denote the diagonal degree matrix whose diagonal entries correspond to the degrees of the vertices. The Laplacian matrix is defined as , while the signless Laplacian matrix is defined as . In addition, the distance matrix is defined such that denotes the length of the shortest path between vertices and in G.

From a chemical perspective, the adjacency matrix reflects local bonding relationships between atoms, the Laplacian and signless Laplacian matrices encode both bonding patterns and vertex connectivity, and the distance matrix captures global molecular size and topological spread. Spectral descriptors derived from these matrices are therefore sensitive to variations in molecular topology and overall structural organization^21,24,26,27.

Eigenvalue based spectral descriptors

Let denote the eigenvalues of the adjacency matrix A(G), the eigenvalues of the Laplacian matrix L(G), the eigenvalues of the signless Laplacian matrix Q(G), and the eigenvalues of the distance matrix . These eigenvalue spectra encode both local and global structural information of the molecular graph. The eigenvalue based spectral descriptors considered in this study are derived from the spectra of graph associated matrices and include indices related to graph energy, Estrada type measures, and Laplacian spectral properties^{13,21,22,26,27}. The descriptors are defined as follows.

1. Adjacency Energy (AE)⁹:

   
2. Adjacency Spectral Radius (ASR)³⁰:

   
3. Adjacency Estrada Index (AEI)⁷:

   
4. Laplacian Energy (LE)¹⁷:

   
5. Laplacian Estrada Index (LEI)^36,37:

   
6. Algebraic Connectivity (AC)¹⁴:

   
7. Signless Laplacian Energy (SLE)^1,15:

   
8. Signless Laplacian Estrada Index (SLEI)^8,16,25:

   
9. Distance Energy (DE)²²:

   

Illustrative calculation

To illustrate the computation of eigenvalue-based spectral descriptors, the molecular structure of Amoxicillin is considered as a representative example. The chemical structure is converted into a molecular graph by representing atoms as vertices and covalent bonds as edges. A schematic representation of the corresponding molecular graph is shown in Fig. 2.

Fig. 2.

Molecular graph representation of Amoxicillin, where vertices represent atoms and edges represent covalent bonds.

Based on this graph representation, the adjacency matrix A(G) is constructed as

where if vertices and are adjacent, and otherwise. Similar procedures are followed to construct the Laplacian, signless Laplacian, and distance matrices.

The eigenvalues of each graph associated matrix are computed numerically using standard linear algebra algorithms. If denote the eigenvalues of the adjacency matrix, the corresponding adjacency energy is calculated as

Analogous procedures are used to compute the remaining spectral descriptors. These descriptors are subsequently employed as molecular variables in the QSPR regression models.

Methodology

This section describes the computational and statistical procedures employed to establish quantitative structure property relationship (QSPR) models for the selected -lactam antibiotics. The methodology consists of molecular graph construction, computation of eigenvalue based spectral descriptors, and subsequent non linear regression analysis.

Experimental physicochemical property data for the selected -lactam antibiotics were obtained from the ChemSpider database maintained by the Royal Society of Chemistry¹⁸. The considered properties include boiling point, molar volume, molar refractivity, polar surface area, polarizability, and surface tension, which are summarized in Table 1.

Table 1.

Physical properties of selected -lactam antibiotics.

Drug	BP	E	FP	MR	PSA	P	ST	MV
	(K)	(kJ/mol)	(K)		(Å)		(mN/m)	(cm/mol)
Amoxicillin	743.2	113.7	403.3	91.5	158	36.3	85.3	236.2
Oxacillin	686.8	105.8	369.2	101.3	138	40.2	77.3	268.5
Cefuroxime	731.7	112.1	396.3	96.7	204	38.3	78.1	241.0
Meropenem	627.4	106.4	333.2	96.8	135	38.4	68.4	268.9
Imipenem	530.2	92.7	274.5	72.7	139	28.8	71.0	183.9
Ertapenem	813.9	124.0	446.0	118.3	182	46.9	84.9	306.2
Clavulanic acid	545.8	94.8	283.9	43.6	87	17.3	82.3	120.3

For each antibiotic, the molecular structure was represented as a simple and connected molecular graph by associating atoms with vertices and covalent bonds with edges. Based on this graph representation, the adjacency, Laplacian, signless Laplacian, and distance matrices were constructed. Eigenvalues of these matrices were computed using standard numerical linear algebra routines implemented in the Python programming environment. The resulting eigenvalue spectra were used to evaluate all spectral descriptors defined in Section "Eigenvalue based topological indices", which are summarized in Table 2.

Table 2.

Graph spectral indices of selected -lactam antibiotics.

Drug	AE	ASR	AEI	LE	LEI	AC	SLE	SLEI	DE
Amoxicillin	31.4464	2.6956	66.2831	40.5900	1039.0585	0.0363	40.8145	1045.2213	266.1190
Oxacillin	36.1719	2.6966	74.9673	44.5195	1155.0329	0.0318	45.2811	1164.1761	308.5235
Cefuroxime	38.0704	2.7025	77.7268	47.0116	1156.4233	0.0320	46.9429	1158.2981	355.6795
Meropenem	32.6042	2.7098	68.6695	41.9118	1029.9828	0.0395	42.2136	1038.3276	273.8790
Imipenem	25.1031	2.6768	52.2732	31.4014	748.7542	0.0609	31.4899	753.6672	185.6230
Ertapenem	41.9261	2.7098	86.5756	52.0314	1231.2364	0.0208	52.3792	1239.5810	448.0094
Clavulanic acid	17.7780	2.6180	38.1042	22.8945	572.4953	0.1659	23.0140	577.3645	92.8351

The computed spectral descriptors served as independent variables in the QSPR models, while the experimental physicochemical properties served as dependent variables. To examine non linear relationships between spectral descriptors and physicochemical properties, three regression models were considered quadratic, logarithmic, and power regression. These models were selected to capture non linear trends commonly observed in QSPR studies involving spectral and topological descriptors.

The quadratic regression model is expressed as

1

the logarithmic regression model is given by

2

and the power regression model is defined as

3

where T denotes the physicochemical property, X represents the spectral descriptor, and R, , and are regression constants.

All regression analyses and statistical evaluations were carried out using SPSS software. For each regression model, the correlation coefficient (R), coefficient of determination (), F-statistic, and corresponding p-values were computed to assess the strength, goodness of fit, and statistical significance of the structure property relationships^33,34.

The predictive accuracy of the regression models was quantified using the root mean square error (RMSE), defined as

4

where and denote the experimental and predicted values of the physicochemical property, respectively, and n denotes the number of data points. The same statistical measures were applied consistently across all regression models to enable direct comparison of their performance.

Results

This section presents the numerical outcomes of the non linear QSPR analysis performed using eigenvalue based spectral descriptors for the selected -lactam antibiotics. Regression results obtained from quadratic, logarithmic, and power models are reported using statistical performance metrics and graphical comparisons.

The regression coefficients and statistical indicators corresponding to the quadratic, logarithmic, and power regression models are summarized in Tables 3,4,5,6,7,8,9,10,11. These tables report the correlation coefficient (R), coefficient of determination (), root mean square error (RMSE), F-statistic, and p-values for each spectral descriptor and physicochemical property. For most properties, all three regression models yield statistically significant correlations between spectral descriptors and experimental data.

Table 3.

Quadratic regression model between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Phy. Pro.	Regression Equation		SE	F-statistic	P-value
BP		0.9128	38.1909	20.9411	0.0076
E		0.8992	4.2432	17.8549	0.0101
FP		0.9127	23.1065	20.9125	0.0076
MR		0.9795	4.2053	95.8913	0.0004
PSA		0.7446	23.2359	5.8309	0.0652
P		0.9793	1.6747	95.0274	0.0004
ST		0.5039	5.6870	2.0319	0.2461
MV		0.9332	19.6239	27.9774	0.0045

Table 4.

Quadratic correlation coefficients between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Index	BP	E	FP	MR	PSA	P	ST	MV
AE	0.8903	0.8861	0.8902	0.9811	0.8425	0.9808	0.5667	0.9556
ASR	0.7534	0.8231	0.7532	0.9544	0.7722	0.9543	0.2919	0.9596
AEI	0.8973	0.8983	0.8971	0.9849	0.8298	0.9847	0.5680	0.9627
LE	0.9071	0.9146	0.9070	0.9845	0.8360	0.9842	0.5710	0.9628
LEI	0.9153	0.8904	0.9153	0.9705	0.8004	0.9706	0.6030	0.9530
AC	0.9554	0.9483	0.9553	0.9897	0.8220	0.9896	0.7099	0.9658
SLE	0.9050	0.9109	0.9049	0.9861	0.8248	0.9859	0.5704	0.9660
SLEI	0.9139	0.8899	0.9138	0.9715	0.7959	0.9716	0.6009	0.9548
DE	0.8963	0.9073	0.8962	0.9821	0.8629	0.9817	0.5639	0.9521

Table 5.

Root mean square error (RMSE) values of quadratic regression model between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Index	BP	E	FP	MR	PSA	P	ST	MV
AE	44.5167	4.6846	26.9258	4.2950	18.7222	1.7155	5.0292	16.9186
ASR	64.2856	5.7394	38.8841	6.6343	22.0809	2.6342	5.8380	16.1464
AEI	43.1508	4.4396	26.1007	3.8459	19.3939	1.5354	5.0239	15.5437
LE	41.1511	4.0859	24.8940	3.9051	19.0687	1.5591	5.0110	15.5238
LEI	39.3688	4.5991	23.8099	5.3587	20.8326	2.1239	4.8693	17.4001
AC	28.8697	3.2076	17.4669	3.1789	19.7929	1.2660	4.2990	14.8851
SLE	41.5888	4.1696	25.1576	3.6898	19.6515	1.4719	5.0133	14.8343
SLEI	39.7001	4.6100	24.0101	5.2702	21.0408	2.0881	4.8790	17.0705
DE	43.3537	4.2482	26.2233	4.1820	17.5647	1.6760	5.0411	17.5523

Table 6.

Logarithmic regression model between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Phy. Pro.	Regression Equation		SE	F-statistic	P-value
BP		0.7345	59.6150	13.8302	0.0137
E		0.7123	6.4140	12.3798	0.0169
FP		0.7344	36.0477	13.8286	0.0197
MR		0.9771	3.9825	213.3004	0.0000
PSA		0.9698	4.5709	160.7184	0.0000
P		0.9766	1.5942	209.1768	0.0000
ST		0.0323	7.1047	0.1668	0.6998
MV		0.9342	17.4276	71.0182	0.0004

Table 7.

Logarithmic correlation coefficients between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Index	BP	E	FP	MR	PSA	P	ST	MV
AE	0.8422	0.8281	0.8422	0.9826	0.8434	0.9823	0.0613	0.9559
ASR	0.6824	0.6980	0.6824	0.9414	0.7718	0.9412	0.1797	0.9316
AEI	0.8484	0.8373	0.8484	0.9863	0.8316	0.9861	0.0680	0.9633
LE	0.8516	0.8440	0.8516	0.9856	0.8362	0.9854	0.0673	0.9633
LEI	0.8570	0.8341	0.8570	0.9724	0.8030	0.9724	0.0924	0.9543
AC	0.8324	0.8224	0.8324	0.9885	0.8258	0.9882	0.0724	0.9600
SLE	0.8504	0.8419	0.8504	0.9872	0.8264	0.9870	0.0671	0.9665
SLEI	0.8564	0.8340	0.8564	0.9733	0.7989	0.9733	0.0918	0.9560
DE	0.8415	0.8352	0.8415	0.9848	0.8622	0.9844	0.0646	0.9534

Table 8.

Root mean square error (RMSE) values of logarithmic regression model between eigenvalue-based topological indices and physical properties of -lactam antibiotics and clavulanic acid.

Index	BP	E	FP	MR	PSA	P	ST	MV
AE	52.7179	5.6662	31.8787	4.1326	18.6717	1.6527	6.0925	16.8709
ASR	71.4694	7.2369	43.2165	7.5016	22.0994	2.9801	6.0046	20.8785
AEI	51.7555	5.5251	31.29712	3.6639	19.30112	1.4646	6.0898	15.4125
LE	51.2495	5.4208	30.9918	3.7592	19.0615	1.5022	6.0901	15.4165
LEI	50.3839	5.5752	30.4658	5.1902	20.7122	2.0576	6.0778	17.1632
AC	54.1805	5.7492	32.7595	3.3658	19.5985	1.3473	6.0880	16.0865
SLE	51.4348	5.4540	31.1035	3.5466	19.5688	1.4160	6.0902	14.7290
SLEI	50.4797	5.5765	30.5238	5.1070	20.9039	2.0239	6.0782	16.8346
DE	52.8194	5.5583	31.9400	3.8631	17.6052	1.5526	6.0912	17.3231

Table 9.

Power regression model between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Phy. Pro.	Regression Equation		SE	F-statistic	P-value
BP		0.7598	0.0865	15.9644	0.0104
E		0.7337	0.0584	13.4174	0.0145
FP		0.7630	0.0975	16.2406	0.0100
MR		0.9768	0.0426	352.0576	0.0000
PSA		0.7434	0.1286	21.9413	0.0054
P		0.9764	0.0429	347.1154	0.0000
ST		0.0327	0.0924	0.1804	0.6886
MV		0.9232	0.0758	98.9532	0.0002

Table 10.

Power correlation coefficients between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Index	BP	E	FP	MR	PSA	P	ST	MV
AE	0.8587	0.8395	0.8607	0.9716	0.8422	0.9713	0.0626	0.9444
ASR	0.6945	0.7090	0.6960	0.9531	0.7667	0.9529	0.1811	0.9512
AEI	0.8650	0.8490	0.8669	0.9775	0.8288	0.9772	0.0694	0.9544
LE	0.8692	0.8566	0.8713	0.9787	0.8361	0.9784	0.0687	0.9561
LEI	0.8717	0.8430	0.8735	0.9676	0.8005	0.9676	0.0941	0.9491
AC	0.8571	0.8398	0.8602	0.9883	0.8207	0.9881	0.0743	0.9609
SLE	0.8676	0.8541	0.8697	0.9812	0.8243	0.9810	0.0685	0.9606
SLEI	0.8710	0.8429	0.8728	0.9688	0.7954	0.9689	0.0934	0.9515
DE	0.8620	0.8501	0.8643	0.9742	0.8622	0.9737	0.0662	0.9426

Table 11.

Root mean square error (RMSE) values of power regression model between eigenvalue-based topological indices and physical properties of -lactam antibiotics.

Index	BP	E	FP	MR	PSA	P	ST	MV
AE	50.3110	5.5033	30.2574	5.4299	18.7687	2.1646	6.0968	19.1034
ASR	70.5779	7.1403	42.6272	6.7527	22.3601	2.6823	6.0081	17.8849
AEI	49.27589	5.3515	29.6255	4.8524	19.4868	1.9336	6.0941	17.3378
LE	48.5894	5.2287	29.1964	4.6956	19.0959	1.8722	6.0944	16.9717
LEI	48.1176	5.4450	28.9328	5.6800	20.8756	2.2514	6.0817	18.1686
AC	50.7840	5.5093	30.4690	3.4704	19.8941	1.3863	6.0920	15.9638
SLE	48.8461	5.2688	29.3562	4.4203	19.7135	1.7611	6.0945	16.0876
SLEI	48.2422	5.4472	29.0103	5.5702	21.1121	2.2070	6.0821	17.7497
DE	49.8638	5.3391	29.9503	5.2095	17.6295	2.0822	6.0954	19.4127

Predicted physicochemical property values obtained from the regression models are presented in Tables 12,13,14. These tables provide a numerical comparison between experimental and predicted values for the considered properties.

Table 12.

Predicted values of physical properties using the quadratic regression model for -lactam antibiotics.

Compound	BP	E	FP	MR	PSA	P	ST	MV
	(K)	(kJ/mol)	(K)		(Å)		(mN/m)	(cm/mol)
Amoxicillin	685.70	108.78	368.50	95.99	151.03	38.06	77.21	241.43
Oxacillin	721.19	112.46	389.96	101.31	162.13	40.17	79.06	263.53
Cefuroxime	719.18	112.25	388.74	101.01	172.71	40.05	78.96	271.21
Meropenem	661.86	106.31	354.09	92.38	153.17	36.61	75.99	248.60
Imipenem	530.73	92.72	274.81	71.49	125.82	28.33	69.66	186.21
Ertapenem	814.60	122.15	446.43	115.00	188.05	45.60	84.04	294.53
Clavulanic acid	545.72	94.81	283.85	43.68	90.03	17.33	82.35	119.45

Table 13.

Predicted values of physical properties using the logarithmic regression model for -lactam antibiotics.

Compound	BP	E	FP	MR	PSA	P	ST	MV
	(K)	(kJ/mol)	(K)		(Å)		(mN/m)	(cm/mol)
Amoxicillin	693.81	108.52	373.40	95.30	91.50	37.79	77.87	241.43
Oxacillin	727.99	111.55	394.07	100.08	98.28	39.68	77.83	263.53
Cefuroxime	728.38	113.34	394.31	99.80	104.82	39.57	77.62	271.21
Meropenem	690.97	109.57	371.68	92.27	92.81	36.59	77.35	248.60
Imipenem	587.95	100.09	309.40	76.95	74.94	30.51	78.55	186.21
Ertapenem	748.63	116.67	406.55	115.10	115.42	45.63	77.35	294.53
Clavulanic acid	501.24	89.73	256.97	41.38	43.11	16.41	80.71	119.45

Table 14.

Predicted values of physical properties using the power regression model for -lactam antibiotics.

Compound	BP	E	FP	MR	PSA	P	ST	MV
	(K)	(kJ/mol)	(K)		(Å)		(mN/m)	(cm/mol)
Amoxicillin	687.77	108.07	369.26	93.27	148.85	36.98	77.61	243.45
Oxacillin	725.42	111.22	392.33	99.62	159.75	39.49	77.57	259.02
Cefuroxime	725.86	113.12	392.59	99.23	170.98	39.34	77.34	258.07
Meropenem	684.74	109.15	367.42	89.47	150.92	35.48	77.07	234.07
Imipenem	583.16	99.77	306.10	72.47	125.32	28.74	78.33	191.91
Ertapenem	749.14	116.76	406.94	122.47	190.91	48.55	77.07	314.70
Clavulanic acid	509.43	90.42	262.51	44.43	90.01	17.62	80.65	121.01

The corresponding predicted versus experimental plots for the regression models are presented in Figures 3,4,5,6,7,8,9,10,11,12,13,14,15,16. The graphical trends observed in these figures are consistent with the numerical values reported in Tables 12,13,14.

Fig. 3.

Visual plot of correlation on quadratic and logarithmic for properties of -lactum antibiotics.

Fig. 4.

Visual plot of correlation on power for properties of -lactum antibiotics.

Fig. 5.

Quadratic regression plots on boiling point and enthalpy.

Fig. 6.

Quadratic regression plots on flash point and molar refractivity.

Fig. 7.

Quadratic regression plots on polar surface area and polarizability.

Fig. 8.

Quadratic regression plots on surface tension and molar volume.

Fig. 9.

Logarithmic regression plots on boiling point and enthalpy.

Fig. 10.

Logarithmic regression plots on flash point and molar refractivity.

Fig. 11.

Logarithmic regression plots on polar surface area and polarizability.

Fig. 12.

Logarithmic regression plots on surface tension and molar volume.

Fig. 13.

Power regression plots on boiling point and enthalpy.

Fig. 14.

Power regression plots on flash point and molar refractivity.

Fig. 15.

Power regression plots on polar surface area and polarizability.

Fig. 16.

Power regression plots on surface tension and molar volume.

Discussion

The results presented in Section “Results” demonstrate that eigenvalue based spectral descriptors capture meaningful structure property relationships for the selected -lactam antibiotics. The observed trends are interpreted below in terms of descriptor behavior, regression model characteristics, and chemical considerations.

Across the examined regression models, quadratic, logarithmic, and power forms exhibit comparable predictive behavior. The slightly improved performance observed for quadratic regression suggests that eigenvalue based spectral descriptors encode non linear dependencies that are more effectively captured by polynomial relationships. Recent QSPR investigations have also explored machine learning frameworks, including neural networks and ensemble methods, to enhance predictive accuracy for drug related physicochemical properties however, such approaches typically require larger datasets and often sacrifice interpretability compared to regression based spectral descriptor models^{2–6,23,28,31,35}.

An examination of descriptor specific behavior indicates that algebraic connectivity, signless Laplacian energy, and distance energy frequently emerge as influential predictors. These descriptors reflect complementary aspects of molecular topology, including global connectivity, degree distribution, and molecular size, which are known to influence physicochemical properties within structurally diverse -lactam antibiotics.

The predicted versus experimental plots shown in Figures 3,4,5,6,7,8,9,10,11,12,13,14,15,16 provide visual support for the numerical regression results. Properties such as boiling point, molar volume, and molar refractivity exhibit close agreement between predicted and experimental values, whereas increased dispersion is observed for polar surface area and surface tension.

The reduced predictive performance observed for polar surface area and surface tension can be attributed to their dependence on electronic distribution, hydrogen bonding, and intermolecular interactions, which are not explicitly encoded in purely topological descriptors. This behavior reflects an intrinsic limitation of graph based representations rather than a limitation of the regression methodology.

Overall, the discussion highlights the suitability of eigenvalue based spectral descriptors for modeling physicochemical properties governed primarily by molecular size, connectivity, and global structural organization in -lactam antibiotics.

Conclusion

In this work, a quantitative structure property relationship (QSPR) analysis was conducted for selected -lactam antibiotics using eigenvalue based spectral descriptors derived from molecular graph representations. Spectral indices obtained from adjacency, Laplacian, signless Laplacian, and distance matrices were combined with non linear regression models to examine relationships between molecular topology and key physicochemical properties. The analysis demonstrates that eigenvalue based spectral descriptors provide meaningful correlations with several physicochemical properties of -lactam antibiotics, particularly those governed by molecular size, connectivity, and global structural organization. Among the considered regression forms, quadratic models consistently exhibit slightly improved predictive performance compared to logarithmic and power models, indicating their suitability for capturing non linear structure property relationships encoded by spectral descriptors. These observations highlight both the strengths and limitations of eigenvalue based spectral descriptors within the context of QSPR modeling. Overall, the findings confirm the applicability of spectral graph descriptors as interpretable and mathematically grounded tools for modeling physicochemical properties of -lactam antibiotics. The study provides a structured basis for future investigations incorporating larger datasets and complementary descriptor classes to further enhance predictive performance.

Author contributions

All authors contributed equally.

Funding

No funding was available for this study.

Data availability

The data used to support the findings of this study are cited at relevant places within the text as references.

Declarations

Competing interests

The authors declare no competing interests.