A (p, q) graph G is called edge-magic if there exists a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , p + q} such that f(u) + f(v) + f(uv) is constant for any edge uv of G. Moreover, G is said to be super edgemagic if f(V (G)) = {1, 2, . . . , p}. Every super edge-magic (p, q) graph is harmonious, sequential and felicitous whenever it is a tree or satisfies q ≥ p. In this paper, we prove...

Let n ? 4 . The helm graph H on 2 ? 1 vertices is obtained from the wheel W by adjoining a pendant edge to each vertex of outer cycle Suppose even. D : = [ d i j ] be distance matrix In this paper, we first show that det ( ) 3 Next, find L and vector u such + ? We also prove an interlacing property between eigenvalues

The purpose of this paper is to analyze the Moore-Penrose pseudo-inversion symmetric real matrices with application in graph theory. We introduce a novel concept positively and negatively pseudo-inverse graphs. also give sufficient conditions on elements block matrix yielding an explicit form its pseudo-inversion. Using we can construct graphs for class which are constructed from original by ad...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

A connected graph G = (V, E) is called a quasi-tree graph if there exists a vertex u0 ∈ V (G) such that G−u0 is a tree. A connected graph G = (V, E) is called a quasi-unicyclic graph if there exists a vertex u0 ∈ V (G) such that G− u0 is a unicyclic graph. Set T (n, k) := {G : G is a n-vertex quasi-tree graph with k pendant vertices}, and T (n, d0, k) := {G : G ∈ T (n, k) and there is a vertex ...

The transmission of a vertex v graph G is the sum distances from to all other vertices in . A irregular if its have pairwise different transmissions. starlike tree T ( k 1 , … t ) obtained by attaching an isolated pendant paths lengths respectively. It proved that + ? 2 odd order, then it irregular. ? 3 and only ? { r : } Additional infinite families among trees bi-starlike are determined. Tran...

Modern network systems (unmanned aerial vehicles groups, social networks, production chains, transport and logistics communication cryptocurrency networks) are distinguished by their multielement nature the dynamics of connections between elements. A number discrete problems on construction optimal substructures described in form various classes graphs NP-complete problems. In this case, variab...

A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of the product degrees adjacent vertices G. In this paper, we introduce several transformations that are useful tools for study extremal properties index. Using these and symmetric structural representations some graphs, determine graphs having maximal with prescri...

We investigate a representation of the automorphism group connected graph X in unimodular matrices Uβ dimension β, where β is Betti number X. classify graphs for which does not embed into Uβ. It follows that if has no pendant vertices and simple cycle, then faithful AutX acts faithfully on H1(X,Z). The latter statement can be viewed as discrete analogue classical Hurwitz's theorem Riemann surfa...