The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino ...

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is local antimagic if vertex-weights are distinct for pair neighboring vertices, where vertex-weight sum labels all edges incident with that vertex. Any induces proper vertex coloring G color its vertex-weight. This naturally leads to concept chromatic number. The number defined be minimum colors take...

In this paper we survey the behavior of various graph operations on the graph parameters clique-width and NLC-width. We give upper and lower bounds for the clique-width and NLC-width of the modi ed graphs in terms of the clique-width and NLC-width of the involved graphs. Therefor we consider the binary graph operations join, co-join, sum, di erence, products, corona, substitution, and 1-sum, an...

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

An injective function f : V pGq Ñ t0, 1, 2, . . . , qu is an odd sum labeling if the induced edge labeling f defined by f puvq fpuq fpvq, for all uv P EpGq, is bijective and f pEpGqq t1, 3, 5, . . . , 2q 1u. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we study the odd sum property of graphs obtained by duplicating any edge of some graphs.

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...

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

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...

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...

In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $delta ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we ...