Changes in the Seidel energy of blow-up graphs under edge deletion

Abstract

Let $S(G)$ denote the Seidel matrix of a simple graph $G$, and let $E_S(G)$ be the Seidel energy of $G$, defined as the sum of the absolute values of the eigenvalues of $S(G)$. In this paper, we study the change of Seidel energy under edge deletion. For an independent-set blow-up graph $G=H[n_1,\ldots,n_p]$, we establish a general structural criterion within the framework of independent-set blow-up graphs. More precisely, if the endpoints of the deleted edge $e$ belong to blow-up parts of sizes $n_a$ and $n_b$, respectively, then $E_S(G-e)>E_S(G)$ whenever both $n_a,n_b$ are at least $4$, or one is $3$ and the other is at least $6$, or one is $2$ and the other is at least $15$. As applications, we obtain the following consequences. First, for every Turán graph $T(n,r)$ with $r\geq4$ and $n\geq4r$, deleting any edge strictly increases the Seidel energy. Second, for complete multipartite graphs, we derive an exact reduced-order spectral criterion for the remaining cases not covered by the structural result. This criterion determines whether the Seidel energy increases, decreases, or remains unchanged after deleting an edge, by using matrices whose orders depend only on the number of partite sets. These results provide affirmative answers to two problems proposed by Tian et al. [\textit{Linear and Multilinear Algebra} 70 (19) (2022), 4597--4614].