Consider a simple graph G = (V,E) and its proper edge colouring c with the elements of the set {1, 2, . . . , k}. The colouring c is said to be neighbour sum distinguishing if for every pair of vertices u, v adjacent in G, the sum of colours of the edges incident with u is distinct from the corresponding sum for v. The smallest integer k for which such colouring exists is known as the neighbour...

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation l of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from {1, 2, . . . , k} denoted it by −→ χu(G). We have 2∆...

Let M be a class of matroids closed under minors and isomorphism. Let M be a 3-connected matroid in M with an N-minor and let N have an exact k-separation (A, B). If there exists a k-separation (X ,Y ) of M such that A ⊆ X and B ⊆ Y , we say the k-separation (A, B) of N is induced in M. In this paper we give new sufficient conditions to determine if an exact k-separation of N is induced inM.

Let N0 be the set of all non-negative integers, let X N0 and P(X) be the the power set of X. An integer additive set-labeling (IASL) of a graph G is an injective function f : V (G) ! P(N0) such that the induced function f+ : E(G) ! P(N0) is defined by f+(uv) = f(u) + f(v), where f(u) + f(v) is the sum set of f(u) and f(v). An IASL f is said to be an integer additive set-indexer (IASI) of a grap...

An edge irregular total k-labeling of a graph G = (V,E) is a labeling φ : V ∪ E → {1, 2, . . . , k} such that the total edge-weights wt(xy) = φ(x) + φ(xy) + φ(y) are different for all pairs of distinct edges. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G. In this paper, we determined the exact values of the total e...

An (edge-)weighted graph is a graph in which each edge e is assigned a nonnegative real number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges, and an optimal cycle is one of maximum weight. The weighted degree w(v) of a vertex v is the sum of the weights of the edges incident with v. The following weighted analogue (and generalization) of a well-known...

Let γ : E(G) −→ N∗ be an edge colouring of a graph G and σγ : V (G) −→ N∗ the vertex colouring given by σγ(v) = ∑ e3v γ(e) for every v ∈ V (G). A neighbour-sumdistinguishing edge-colouring of G is an edge colouring γ such that for every edge uv in G, σγ(u) 6= σγ(v). The study of neighbour-sum-distinguishing edge-colouring of graphs was initiated by Karoński, Łuczak and Thomason [8]. They conjec...

In this paper, we study the total edge irregularity strength of some well known graphs. An edge irregular total k-labeling φ : V ∪E → {1, 2, . . . , k} of a graph G = (V,E) is a labeling of vertices and edges of G in such a way that for any different edges xy and x′y′ their weights are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which G has an edge irre...

In this paper, we consider security aspects of network routing in a game-theoretic framework where an attacker is empowered with an ability for intrusion into edges of the network; on the other hand, the goal of designer is to choose routing paths. We interpret the secure routing problem as a two player zero sum game. The attacker can choose one or more edges for intrusion, while the designer h...

For a simple path Pr on r vertices, the square of Pr is the graph P2 r on the same set of vertices of Pr , and where every pair of vertices of distance two or less in Pr is connected by an edge. Given a (p, q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k, k + 1, . . . , k + q − 1, so that the induc...