‎in this paper we introduce mixed unitary cayley graph $m_{n}$‎ ‎$(n>1)$ and compute its eigenvalues‎. ‎we also compute the energy of‎ ‎$m_{n}$ for some $n$‎.

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

Let G be a graph and t : V ( G ) , {0, 1}, where t(V(G)) is even. (If X~_ V(G), then t ( X ) : = E { t (x):xeX}. ) A t-join is a set F ~ E ( G ) with d~(x)=t(x) (mod 2), Vx e V(G). (dF(x) denotes the number of edges of F incident with x, where loops count twice.) t-joins contain Chinese postman tours, matchings and minimum weight paths as a special case. (el. [1, 7]). If X c V(G), let 6 ( X ) =...

We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the...

On-line multicasting has received a lot of attention lately, because of its great practical importance. However, most research assumes undirected graphs, which fail to model the directivity of real networks. In this paper, we ll this gap by ooering upper and lower bounds for the on-line directed multicast problem in a worst case analysis. We examine two types of on-line problems: the join-only ...

The vertex distance complement (VDC) matrix \(\textit{C}\), of a connected graph \(G\) with set consisting \(n\) vertices, is real symmetric \([c_{ij}]\) that takes the value \(n - d_{ij}\) where \(d_{ij}\) between vertices \(v_i\) and \(v_j\) for \(i \neq j\) 0 otherwise. spectrum subdivision join, \(G_1 \dot{\bigvee} G_2\) edge join \underline{\bigvee} regular graphs \(G_1\) \(G_2\) in terms ...

Fix d ≥ 2, and let X be either Z or the points of a Poisson process in R of intensity 1. Given parameters r and p, join each pair of points of X within distance r independently with probability p. This is the simplest case of a ‘spread-out’ percolation model studied by Penrose [8], who showed that, as r → ∞, the average degree of the corresponding random graph at the percolation threshold tends...