ON SPECTRUM OF I-GRAPHS AND ITS ORDERING WITH RESPECT TO SPECTRAL MOMENTS

Abstract

Suppose $G$ is a graph, $A(G)$ its adjacency matrix, and $μ_1(G), μ_2(G), \cdots, μ_n(G)$ are eigenvalues of $A(G)$. The numbers $S_k(G) = \sum_{i=1}^n μ^k_i(G)$, $0 \leq k \leq n − 1$, are said to be the k−th spectral moment of $G$ and the sequence
$S(G) = (S_0(G), S_1(G), \sdots, S_{n−1}(G))$ is called the spectral moments sequence of $G$. For two graphs $G_1$ and $G_2$, we define $G_1 \leq_S G_2$, if there exists an integer
$k$, $1 \leq k \leq n − 1$, such that for each $i$, $0 \leq i \leq k − 1$, $S_i(G_1) = S_i(G_2)$ and
$S_k(G_1) < S_k(G_2)$.
The I−graph $I(n, j, k)$ is a graph of order $2n$ with the vertex and edge sets
$V(I(n, j, k) = \{u_0, u_1, \cdots, u_{n−1}, v_0, v_1, \cdots, v_{n−1}\}$,
$E(I(n, j, k) = \{u_iu{u+j}, u_iv_i, v_iv_{i+k} ; 0 \leq i \leq n − 1\}$,
respectively. The aim of this paper is to compute the spectrum of an arbitrary
I−graph and the extremal I−graphs with respect to the S−order.