1*PG. Department of Mathematics,
KLE Society’s S.K. Arts and HSK Science College, Hubballi, Karnataka, INDIA.
2Department of Mathematics,
KLE Society’s P.C. Jabin Science College, Hubballi, Karnataka, INDIA.

ABSTRACT

Let 𝑇 be any tree and 𝑃 be set of paths, given a labelling 𝑤:𝐸(𝑇)→𝑍+ of the edges of 𝑇 for a path 𝑝∈𝑃, such that 𝑤(𝑝)=Σ𝑤(𝑒)𝑒∈𝑝. Here each path in 𝑇 has distinct sum weights along its edges and those sums are the consecutive positive integers 1 through (𝑛2). Such labelling of a 𝑇 is a Leech tree labelling. In this article we define a Leech tree matrix ℓ(𝑇), as follows; ℓ(𝑇)=(𝑎𝑖𝑗)={ 1, 𝑖𝑓 𝑤(𝑝) 𝑖𝑠 𝑒𝑣𝑒𝑛−1, 𝑖𝑓 𝑤(𝑝) 𝑖𝑠 𝑜𝑑𝑑0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
and discuss some spectral properties of Leech tree matrix and deduce Leech tree energy 𝐸ℓ(𝑇)=𝐸(𝐾𝑛)=2(𝑛−1).
AMS Subject Classifications: 05C50.

Keywords :Leech tree, Leech tree matrix, Leech tree spectrum and Leech tree energy.