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The second immanantal polynomials of Laplacian matrices of unicyclic graphs

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  • Wu, Tingzeng
  • Yu, Yong
  • Gao, Xing

Abstract

Let M=(aij) be a square matrix of order n. The second immanant of M isd2(M)=∑σ∈Snχ(σ)∏t=1natσ(t),where χ is the irreducible character of Sn corresponding to the partition (2,1n−2). Let G be a graph. Denote by L(G) the laplacian matrix of G. The polynomial ϕ(G,x)=d2(xI−L(G))=∑k=0n(−1)kck(G)xn−k is called the second immanantal polynomial of G, where ck(G) is the k-th coeficient of ϕ(G,x). The coefficient ck(G) admit various algebraic and topological interpretations for a graph. Merris first considered the coefficient cn−1(G), and he proposed an open problem: given n and m, can the graph G be determined for which cn−1(G) is a minimum or maximum? In this article, we investigate the problem, and we construct six graph operations each of which changes the value of cn−1(G). Using these graph operations, we determine the bound of cn−1(G) of unicyclic graphs. In particular, we also determined the corresponding extremal graphs.

Suggested Citation

  • Wu, Tingzeng & Yu, Yong & Gao, Xing, 2023. "The second immanantal polynomials of Laplacian matrices of unicyclic graphs," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    • Handle: RePEc:eee:apmaco:v:451:y:2023:i:c:s0096300323002072
    DOI: 10.1016/j.amc.2023.128038
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    References listed on IDEAS

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    1. Yu, Guihai & Qu, Hui, 2018. "The coefficients of the immanantal polynomial," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 38-44.
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