An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [5] conjectured th...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

Abstract. The energy of a simple graph G arising in chemical physics, denoted by E(G), is defined as the sum of the absolute values of eigenvalues of G. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics , to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free...

For a finite group G, the proper power graph P∗(G) of G is the graph whose vertices are non-trivial elements of G and two vertices u and v are adjacent if and only if u 6= v and um = v or vm = u for some positive integer m. In this paper, we consider the complement of P∗(G), denoted by P(G). We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a ...

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . ,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [4] conjectured tha...

................................................................................................................................................. 3 Acknowledgements................................................................................................................................ 6 List of Tables...........................................................................................

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane as vertices and an adjacency resembling that of star Cayley graphs. The graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copi...

Let K be a class of graphs. Then, K is said to have a finite duality if there exists a pair (F , U), where U ∈ K and F is a finite set of graphs, such that for any graph G in K we have G ≤ U if and only if F 6≤ G for all F ∈ F (“ ≤ ” is the homomorphism order). We prove that the class of planar graphs has no finite duality except for two trivial cases. We also prove that a 5-colorable toroidal ...