Some Irregular Total Labelings of Expansion Graphs expan(Pm, Cn)

For a simple graph G = (V(G), E(G)) with vertex set V(G) and edge set E(G), a total labeling lambda : V(G) boolean OR E(G) -> {1, 2, ..., k} is called an edge-irregular total k-labeling of G if for any two different edges e = e(1)e(2) and f = f(1)f(2) in E(G), we have wt(e) not equal wt( f), where wt(e) = lambda(e(1)) + lambda(e) + lambda(e(2)). Meanwhile, a total labeling theta : V(G) boolean OR E(G) -> {1, 2,..., k} is called a vertex-irregular total k-labeling of G if for any two different vertices u and v in V(G), we obtain wt(u) not equal wt(v), where wt(u) = theta(u) + Sigma(uv is an element of E(G)) theta(uv). The minimum value of k for which there exists an edge (a vertex)-irregular total k-labeling of G is called the total edge (vertex) irregular strength of G, denoted by tes(G) (tvs(G)). In this paper, we consider an expansion graph expan (P-m, C-n), where P-m is a path on m vertices and C-n is a cycle on n vertices. An expan (P-m, C-n) is a graph obtained from a copy of P-m and m + n copies of C-n by sticking the i-th copy of C-n at i-th vertex of P-m and sticking the j-th copy of C-n at the j-th edge of P-m. We determine tes(expan(P-m, C-n)) and tvs(expan(P-m, C-n)) for any integers m >= 2 and n >= 3.