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In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds. [ABSTRACT FROM AUTHOR]

For a simple graph with vertex set { v 1 , v 2 , ... , v n } with degree sequence d v i of vertex v i , i = 1 , 2 , ... , n , the inverse sum indeg matrix ( I S I -matrix) A I S I (G) = (a i j) n × n of G is defined by a i j = d v i d v j d v i + d v j , if v i is adjacent to v j , and zero, otherwise. The multiset of eigenvalues of A I S I (G) is the I S I -spectrum of G and the sum of their absolute values is the I S I -energy of G . In this paper, we modify the two results of (Li, Ye and Broersma, 2022), give the correct characterization of the extremal graphs and thereby obtain better bounds than the already known results. Moreover, we also discuss the QSPR analysis and carry the statistical modelling (linear, logarithmic and quadratic) of the physicochemical properties of anticancer drugs with the I S I -index (energy). [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the vertex set V = { v 1 , ... , v n } and denote by d v i the degree of the vertex v i . The modified Sombor index of G is the addition of the numbers (d v i 2 + d v j 2) − 1 / 2 over all of the edges v i v j of G. The modified Sombor matrix A M S (G) of G is the n by n matrix such that its (i , j) -entry is equal to (d v i 2 + d v j 2) − 1 / 2 when v i and v j are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of A M S (G) . The sum of the absolute eigenvalues of A M S (G) is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is 2 ; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven. [ABSTRACT FROM AUTHOR]