Let G=(V(G),E(G)) be a connected simple undirected graph with non empty vertex set V(G) and edge set E(G). For a positive integer k, by an edge irregular total k-labeling we mean a function f : V(G)UE(G) --> {1,2,...,k} such that for each two edges ab and cd, it follows that f(a)+f(ab)+f(b) is different from f(c)+f(cd)+f(d), i.e. every two edges have distinct weights. The minimum k for which G ...

We investigate the irregularity strength (s(G)) and total vertex irregularity strength (tvs(G)) of circulant graphs Cin(1, 2, . . . , k) and prove that tvs(Cin(1, 2, . . . , k)) = ⌈ n+2k 2k+1 ⌉ , while s(Cin(1, 2, . . . , k)) = ⌈ n+2k−1 2k ⌉ except if either n = 2k + 1 or if k is odd and n ≡ 2k + 1(mod4k), then s(Cin(1, 2, . . . , k)) = ⌈ n+2k−1 2k ⌉ + 1.

<p>Foster (1932) performed a mathematical census for all connected symmetric cubic (trivalent) graphs of order <em>n</em> with <em>n </em>≤ 512. This then was continued by Conder et al. (2006) and they obtained the complete list ≤ 768. In this paper, we determine total vertex irregularity strength such Foster. As result, values strengths from Foster strengthen conj...

We investigate the following modification of the well known irregularity strength of graphs. Given a total weighting w of a graph G = (V,E) with elements of a set {1, 2, . . . , s}, denote wtG(v) = ∑ e3v w(e)+w(v) for each v ∈ V . The smallest s for which exists such a weighting with wtG(u) 6= wtG(v) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of...

For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ‎connected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.