Seidel polynomials of an edge embedded $ t $-partite Turán graphs and Seidel energy analysis for the $ 4 $-partite case

Masood Ur Rehman; Anjie Yang; Muhammad Ajmal; B. G. Rodrigues; Ayse Dilek Maden; Gaixiang Cai; Masood Ur Rehman; Anjie Yang; Muhammad Ajmal; B. G. Rodrigues; Ayse Dilek Maden; Gaixiang Cai

doi:10.3934/math.20261279

AIMS Mathematics

2026, Volume 11, Issue 10: 32514-32537. doi: 10.3934/math.20261279

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Research article

Seidel polynomials of an edge embedded $ t $-partite Turán graphs and Seidel energy analysis for the $ 4 $-partite case

1.
School of Mathematics and Statistics, Anqing Normal University, Anqing 246133, China
2.
Department of Mathematics, Shanghai University, Shanghai 200444, China
3.
Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0028, South Africa
4.
Department of Mathematics, Faculty of Science, Selcuk University, Konya 42130, Turkiye

Received: 20 July 2026 Revised: 24 September 2026 Accepted: 30 September 2026 Published: 09 October 2026
MSC : 05C50, 05C38, 05C82

For a simple graph $T$ of order $n$, the Seidel matrix is defined by $S(T) = J - I - 2A(T)$, where $J$ is the all-ones matrix, $I$ is the identity matrix, and $A(T)$ is the adjacency matrix. The characteristic polynomial of $S(T)$ is called the Seidel polynomial, and the sum of the absolute values of its eigenvalues is the Seidel energy of $T$. In this paper, we construct explicit Seidel polynomials for $t$-partite Turán graphs $T(n, t)$ after embedding an edge between two non-adjacent vertices. Next, we analyze the effect on the Seidel energy of the $ 4 $-partite Turán graph $ T(n, 4) $, extending the previous results for bipartite and tripartite Turán graphs. We prove that for $ n \geq 12 $, the Seidel energy strictly increases under edge embedding in the $4$-partite case. Meanwhile, we also prove that for $ t \geq 3 $, the Seidel energy of $ t $-partite Turán graph $ T(2t, t)\cong K_{2, 2, \ldots, 2} $ strictly decreases under edge embedding. The proofs rely on block determinant techniques, coefficient relations of characteristic polynomials, and explicit root bounds obtained via the intermediate value theorem. These results contribute to the broader understanding of extremal properties of Seidel energy and provide new evidence toward the general monotonicity question posed in recent literature.
- $ t $-partite Turán graph,
- edge embedding,
- Seidel polynomial,
- Seidel energy
Citation: Masood Ur Rehman, Anjie Yang, Muhammad Ajmal, B. G. Rodrigues, Ayse Dilek Maden, Gaixiang Cai. Seidel polynomials of an edge embedded $ t $-partite Turán graphs and Seidel energy analysis for the $ 4 $-partite case[J]. AIMS Mathematics, 2026, 11(10): 32514-32537. doi: 10.3934/math.20261279

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Abstract

For a simple graph $T$ of order $n$, the Seidel matrix is defined by $S(T) = J - I - 2A(T)$, where $J$ is the all-ones matrix, $I$ is the identity matrix, and $A(T)$ is the adjacency matrix. The characteristic polynomial of $S(T)$ is called the Seidel polynomial, and the sum of the absolute values of its eigenvalues is the Seidel energy of $T$. In this paper, we construct explicit Seidel polynomials for $t$-partite Turán graphs $T(n, t)$ after embedding an edge between two non-adjacent vertices. Next, we analyze the effect on the Seidel energy of the $ 4 $-partite Turán graph $ T(n, 4) $, extending the previous results for bipartite and tripartite Turán graphs. We prove that for $ n \geq 12 $, the Seidel energy strictly increases under edge embedding in the $4$-partite case. Meanwhile, we also prove that for $ t \geq 3 $, the Seidel energy of $ t $-partite Turán graph $ T(2t, t)\cong K_{2, 2, \ldots, 2} $ strictly decreases under edge embedding. The proofs rely on block determinant techniques, coefficient relations of characteristic polynomials, and explicit root bounds obtained via the intermediate value theorem. These results contribute to the broader understanding of extremal properties of Seidel energy and provide new evidence toward the general monotonicity question posed in recent literature.

References

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