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Trees with integer reciprocal eigenvalue property
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DOI：10.1080/03081087.2026.2666137.png)
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Abstract
En 中文
Let G be a graph and A(G) be the adjacency matrix of G. Let m >= 2 be a positive integer. Then G is said to have the property m-(R) if m lambda is an eigenvalue of G whenever lambda is an eigenvalue of G. Further, if lambda and m lambda have the same multiplicity for each eigenvalue lambda, then we say that G satisfies property m-(SR). In this article, we provide a class of graphs with property m-(SR) using the corona operation. We prove that a tree satisfying property m-(R) must be singular. A result relating the coefficients of the characteristic polynomial of graphs with property m-(SR) is obtained. Using this, we construct classes of trees satisfying property m-(SR). A caterpillar tree Pk(n1,& mldr;,nk) is obtained from a path Pk on vertices 1,& mldr;,k by attaching ni pendant vertices to vertex i, for each i=1,& mldr;,k . Let Pn denote the class of all such caterpillar trees on n vertices, where each ni is a positive integer. Observing that property m-(R) and property m-(SR) are equivalent within the class Pn , we characterize all trees in Pn that satisfy property m-(SR) for prime m >= 7 .
Keywords:
Tree
adjacency matrix
reciprocal eigenvalue property
caterpillar
property m-(SR)
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