We continue the study of graphs defined by a certain adjacency property by investigating the n-existentially closed line-critical graphs. We classify the 1-e.c. line-critical graphs and give examples of 2-e.c. line-critical graphs for all orders ≥ 9.

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph $\mathbb{T}$-gain $\Phi$. Similarly to what happens in context signed graphs, real number $\min Spec(A(\mathcal (\Phi))$, that is, smallest eigenvalue adjacency matrix L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring (\Phi))=-2$ are identified. When such ful...

A radio labeling of a graph G is mapping f : V ( ) → {0, 1, 2,...} such that | u − v ⩾ d + 1 , holds for every pair vertices and where the diameter distance between in . The number denoted by r n smallest t admits with = max ⁡ { ∈ } block each (induced maximal 2-connected subgraph) complete graph. In this paper, lower bound graphs established. which achieves called We prove three necessary suff...

Thomassen conjectured that every 4-connected line graph is hamiltonian. It has been proved that every 4-connected line graph of a claw-free graph, or an almost claw-free graph, or a quasi-claw-free graph, is hamiltonian. In 1998, Ainouche et al. [2] introduced the class of DCT graphs, which properly contains both the almost claw-free graphs and the quasi-claw-free graphs. Recently, Broersma and...

A graph G is collapsible if for every even subset X ~ V(G), G has a subgraph r such that G E(J ') is connected and the set of odd-degree vertices of J' is X. A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G. In this paper. we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between t...

The aim of this work is to investigate the F-index and F-coindex of the line graphs of the cycle graphs, star graphs, tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts. F-index of the line graph of subdivision graph of square grid graph, 2D-lattice, nanotube and nanotorus of TUC4C8[p, q] are also investigated here.