Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph ismaximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximall...

We present explicit formulae for the eccentric connectivity index and Wiener index of 2-dimensional square and comb lattices with open ends. The formulae for these indices of 2-dimensional square lattices with ends closed at themselves are also derived. The index for closed ends case divided by the same index for open ends case in the limit N →&infin defines a novel quantity we call compression...

Introduction: Medial temporal sclerosis (MTS) is a form of mesial temporal lobe epilepsy (mTLE). It is typically characterized by structural alterations in hippocampus (HC) and related mesial temporal lobe (MTL) network. Resting state functional connectivity (RSFC) is considered an ideal technique in quantifying the dysfunction and maladaptation in MTL network. It is well- dem...

Let M and N be internally 4-connected binary matroids such that M has a proper N -minor, and |E(N)| ≥ 7. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N -minor whenever e is in a triangle of M , and M/e has no N -minor whenever e is in a triad of M , then M has a minor, M ′, such that M ′ is internal...

We prove that if N is an internally 4-connected minor of an internally 4-connected binary matroid M with E(N) ≥ 4, then there exist matroids M0,M1, . . . ,Mn such that M0 ∼= N , Mn = M , and, for each i ∈ {1, . . . , i}, Mi−1 is a minor of Mi, |E(Mi−1)| ≥ |E(Mi)| − 2, and Mi is 4-connected up to separators of size 5.

In an earlier paper, we proved that an internally 4connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in that it produces a smaller minor from ...