For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic}  if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$  defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$  the group of integers modulo $k...

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

In the complete graph K2m+3 for m ≥ 2, we study some structures of simple non-isomorphic Hamiltonian sub-graphs of the form H ( 2m+3 , 6m+3) for m ≥ 2 . The various structures of the form H(2m+3, 6m+3) for m ≥ 2 are found and some of them are observed to give some forms of metal atom cluster compound of chemistry . Finally, we establish an algorithm to solve the traveling salesman problem when ...

We build off the game, NimG [7] to create a version named Neighboring Nim. By reducing from Geography, we show that this game is PSPACE-hard. The games created by the reduction share strong similarities with Undirected (Vertex) Geography and regular Nim, both of which are solvable in polynomial-time. We show how to construct PSPACE-complete versions with nim heaps ∗1 and ∗2. This application of...

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

Combining the the results of A.R. Meyer and L.J. Stockmeyer " The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space " , and K.S. Booth " Iso-morphism testing for graphs, semigroups, and finite automata are polynomiamlly equivalent problems " shows that graph isomorphism is PSPACE-complete. The equivalence problem for regular expressions was shown to be PSPACE-...

The two-player game of Nim on graphs is played on a regular graph with positively weighted edges by moving alternately from a fixed starting vertex to an adjacent vertex, decreasing the weight of the incident edge to a strictly smaller non-negative integer. The game ends when a player is unable to move since all edges incident with the vertex from which the player is to move have weight zero. I...

the set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $ns_n$‎. ‎a sequence $piin ns_{n}$ is said to be graphic if it is the degree sequence of a simple graph $g$ on $n$ vertices‎, ‎and such a graph $g$ is called a realization of $pi$‎. ‎the set of all graphic sequences in $ns_{n}$ is denoted by $gs_{n}$‎. ‎the complete product split graph on $l‎ ...

The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph L(G). The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in L(G). Attributed to Gallai (1967), the study ...