in this work, using eigenvalues and eigenvectors of unitary cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allowing pst occurring in its unitary cayley graph. moreover, we have some developments when $r$ is a product of local rings.

Let $G$ be a finite group. The prime graph of $G$ is a graph $Gamma(G)$ with vertex set $pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. In this paper we prove that if $Gamma(G)=Gamma(G_2(5))$, then $G$ has a normal subgroup $N$ such that $pi(N)subseteq{2,3,5}$ and $G/Nequiv G_2(5)$.

Besides breadth first search, which we saw in class in relation to Dijkstra’s algorithm, there is one other fundamental algorithm for searching a graph: depth first search. To better understand the need for these procedures, let us imagine the computer’s view of a graph that has been input into it, in the adjacency list representation. The computer’s view is fundamentally local to a specific ve...

Let 1 (G) : : : n (G) be the eigenvalues of the adjacency matrix of a graph G of order n; and G be the complement of G: Suppose F (G) is a …xed linear combination of i (G) ; n i+1 (G) ; i G ; and n i+1 G ; 1 i k: We show that the limit lim n!1 1 n max fF (G) : v (G) = ng always exists. Moreover, the statement remains true if the maximum is taken over some restricted families like “Kr-free”or “r...

Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem w...

The unitary Cayley graph Xn has vertex set Zn = {0, 1,…, n-1} and vertices u and v are adjacent, if gcd(uv, n) = 1. In [A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881–1889], the energy of unitary Cayley graphs is computed. In this paper the Wiener and hyperWiener index of Xn is computed.