Transactions on Combinatorics
Volume:10 Issue: 4, Dec 2021

‎For a connected graph $G=(V,E)$ of order at least two‎, ‎an edge detour monophonic set of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$‎. ‎The edge detour monophonic number of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$‎. ‎A subset $T$ of $S$ is a forcing edge detour monophonic subset for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$‎. ‎A forcing edge detour monophonic subset for $S$ of minimum cardinality is a minimum forcing edge detour monophonic subset of $S$‎. ‎The forcing edge detour monophonic number $f_{edm}(S)$ in $G$ is the cardinality of a minimum forcing edge detour monophonic subset of $S$‎. ‎The forcing edge detour monophonic number of $G$ is $f_{edm}(G)=min{f_{edm}(S)}$‎, ‎where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$‎. ‎We determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs‎. ‎It is shown that for every pair a‎, ‎b of positive integers with $0leq a<b$ and $bgeq 2$‎, ‎there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$‎.

Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplane and sphere arrangements, polytopes, directed graphs, and even chirality in molecular chemistry. Independently and with another method, Hochst"{a}ttler and Welker also computed in 2019 the same determinant.

We generalize some convolution identities due to Witula and Qi et al‎. ‎involving the central binomial coefficients and Catalan numbers‎. ‎Our formula allows us to establish many new identities involving these important quantities‎, ‎and recovers some known identities in the literature‎. ‎Also‎, ‎we give new proofs of Shapiro's Catalan convolution and a famous identity of Haj'{o}s‎.

The connective eccentricity index (CEI) of a graph $G$ is defined as $xi^{ce}(G)=sum_{v in V(G)}frac{d_G(v)}{varepsilon_G(v)}$, where $d_G(v)$ is the degree of $v$ and $varepsilon_G(v)$ is the eccentricity of $v$. In this paper, we characterize the unique trees with the maximum and minimum CEI among all $n$-vertex trees and $n$-vertex conjugated trees with fixed maximum degree, respectively.

Let $kgeq 1$ be an integer and $mathcal{I}_k$ be‎ ‎the set of all finite groups $G$ such that every bi-Cayley graph $BCay(G,S)$ of $G$ with respect to‎ ‎subset $S$ of length $1leq |S|leq k$ is integral‎. ‎Let $kgeq 3$‎. ‎We prove that a finite group $G$ belongs to $mathcal{I}_k$ if and‎ ‎only if $GcongBbb Z_3$‎, ‎$Bbb Z_2^r$ for some integer $r$‎, ‎or $S_3$‎.