Let $Gamma_{n,kappa}$ be the class of all graphs with $ngeq3$ vertices and $kappageq2$ vertex connectivity. Denote by $Upsilon_{n,beta}$ the family of all connected graphs with $ngeq4$ vertices and matching number $beta$ where $2leqbetaleqlfloorfrac{n}{2}rfloor$. In the classes of graphs $Gamma_{n,kappa}$ and $Upsilon_{n,beta}$, the elements having maximum augmented Zagreb index are determined.

We consider the class F of 2-connected non-planar K3,3-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K5 (the core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. A method to enumerate these graphs in the labelled case is described. Moreover, we enumerate the ho...

In this paper, we determine all connected 5-arc transitive cubic Cayley graphs on the alternating group A47; there are only two such graphs (up to isomorphism). By earlier work of the authors, these are the only two nonnormal connected cubic arc-transitive Cayley graphs for finite nonbelian simple groups, and so this paper completes the classification of such non-normal Cayley graphs.

In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains claw (K1,3) net (a fixed six-vertex graph) as an induced subgraph. This implies subject to being connected, these two are the only minimal (under taking subgraphs) graphs with no path. Brousek (1998) characterized 2-connected, non-Hamiltonian do not contain We characterize 2-connect...

We consider the class F of 2-connected non-planar K3,3-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K5 (the core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. A method to enumerate these graphs in the labelled case is described. Moreover, we enumerate the ho...

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...

Thtte's Wheels Theorem states that a minimally 3-connected non-wheel graph G with at least four vertices contains at least one edge e such that the contraction of e from G produces a graph which is both 3-connected and simple. The edge e is said to be non-essential. We show that a minimally 3-connected graph which is non-planar contains at least six non-essential edges. The wheel graphs are the...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...