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The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops A G S will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix A G S − σ n I , where σ denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph G S is explored. In addition, the existence of a graph with self-loops for every odd energy is proved. [ABSTRACT FROM AUTHOR]

This paper established the idea of minimum evdominating matrix of semigraph and calculated its energy. The minimum ev-dominating energy EmeD(G) of a semigraph G is the sum of the absolute values of the eigenvalues of the minimum ev-dominating matrix. Here some results are also derived in connection with the energy of minimum evdominating matrix. Some lower bounds are also established. [ABSTRACT FROM AUTHOR]

The distance energy of a connected graph G is the sum of absolute values of the eigenvalues of the distance matrix of G. In this paper, we study how the distance energy of the complete split graph G S (m , n) = K m + K ¯ n changes when an edge is deleted from it. The complete split graph G S (m , n) consists of a clique on m vertices and an independent set on n vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph G S (m , n) always increases when an edge is deleted from it. [ABSTRACT FROM AUTHOR]

The spectral radius ρ (A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2 × 2 block matrix A = ( 0 A 12 A 21 A 22 ) , then ρ (A) 2 ≤ ρ 2 2 + θ ⁎ , where θ ⁎ is the largest real root of the equation μ 2 = (x − ν) 2 (ρ 2 2 + x) and ρ 2 = ρ (A 22) , μ = ρ (A 12 A 22 A 21) , ν = ρ (A 12 A 21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. The Seidel energy of G , denoted by E s (G) , is the sum of absolute values of all Seidel eigenvalues of G. A complete r -partite graph with n vertices is called r -partite Turán graph, denoted by T (n , r) , if the number of vertices of each part is either ⌊ n r ⌋ or ⌈ n r ⌉. In 2021, Tian et al. proposed a problem that for any edge e of Turán graph T (n , r) with r ≥ 4 , does there exist a sufficiently large integer n 0 such that E s (T (n , r)) < E s (T (n , r) − e) whenever n ≥ n 0 . In this paper, we prove that the Seidel energy of T (n , 5) of order n ≥ 5 is always increased when an edge is deleted. [ABSTRACT FROM AUTHOR]

Let Γ = (G , σ) be a signed graph. The spectral radius of Γ is the largest absolute value of its adjacency eigenvalues. In this paper we identify the real numbers which are limit points of spectral radii of signed graphs. This is one of the two aspects of a problem in spectral graph theory known as the Hoffman program, implemented here for signed graphs. [ABSTRACT FROM AUTHOR]