let $g = (v, e)$ be a simple graph. denote by $d(g)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$  and  $a(g)$ the adjacency matrix of $g$. the  signless laplacianmatrix of $g$ is $q(g) = d(g) + a(g)$ and the $k-$th signless laplacian spectral moment of  graph $g$ is defined as $t_k(g)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...

We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianisation of a cyclically presented...

A side skirt is a planar rooted tree T, T≠P2, where the root of T vertex degree at least two, and all other vertices except leaves are three. reduced Halin graph or skirted plane G=T∪P, skirt, P path connecting in order determined by embedding T. The structure graphs contains both symmetry asymmetry. For n≥2 Pn=v1v2v3⋯vn as length n−1, we call Cartesian product G Pn, n-generalized prism over G....

We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no “prism” (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class A′ of graphs that contain no odd hole, no antihole of length at least 5, and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are r...

A graph G is called a prism fixer if γ(G×K2) = γ(G), where γ(G) denotes the domination number of G. A symmetric γ-set of G is a minimum dominating set D which admits a partition D = D1 ∪ D2 such that V (G)−N [Di] = Dj , i, j = 1, 2, i 6= j. It is known that G is a prism fixer if and only if G has a symmetric γ-set. Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discus...

Every connected simple graph G has an acyclic orientation. Deene a graph AO(G) whose vertices are the acyclic orientations of G and whose edges join orientations that diier by reversing the direction of a single edge. It was known previously that AO(G) is connected but not necessarily Hamiltonian. However, Squire 3] proved that the square AO(G) 2 is Hamiltonian. We prove the slightly stronger r...

Every connected simple graph G has an acyclic orientation. Define a graph AO(G) whose vertices are the acyclic orientations of G and whose edges join orientations that differ by reversing the direction of a single edge. It was known previously that AO(G) is connected but not necessarily Hamiltonian. However, Squire [3] proved that the square AO(G) is Hamiltonian. We prove the slightly stronger ...

A radio labeling is an assignment c : V (G) → N such that every distinct pair of vertices u, v satisfies the inequality d(u, v) + |c(u) − c(v)| ≥ diam(G) + 1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted Zn,s, s ≥ 1, n ≥ s, have vertex set {(i, j) | i = 1, 2 and j = 1, . . . ...