Let \(G\) be a connected simple graph with girth \(g\) and minimal degree \(\delta \geq 3\). If \(G\) is not up-embeddable, then, when \(G\) is 1-edge connected,

\[\gamma_M(G) \geq \frac{D_1(\delta,g)-2}{2D_1(\delta,g)-1}\beta(G)+ \frac{D_1(\delta,g)+1}{2D_1(\delta,g)-1}.\]

When \(G\) is \(k\)(\(k = 2, 3\))-edge connected ,

\[\gamma_M(G) \geq \frac{D_k(\delta,g)-1}{2D_k(\delta,g)}\beta(G)+ \frac{D_k(\delta,g)+1}{2D_k(\delta,g)}.\]

The functions \(D_k(\delta, g)\) (\(k = 1, 2, 3\)) are increasing functions on \(\delta\) and \(g\).

In this paper, the authors discuss the values of a class of generalized Euler numbers and generalized Bernoulli numbers at rational points.

For two vertices \(u\) and \(v\) in a graph \(G = (V,E)\), the detour distance \(D(u,v)\) is the length of a longest \(u-v\) path in \(G\). A \(u-v\) path of length \(D(u,v)\) is called a \(u-v\) detour. A set \(S \subseteq V\) is called a weak edge detour set if every edge in \(G\) has both its ends in \(S\) or it lies on a detour joining a pair of vertices of \(S\). The weak edge detour number \(dn_w(G)\) of \(G\) is the minimum order of its weak edge detour sets and any weak edge detour set of order \(dn_w(G)\) is a weak edge detour basis of \(G\). Certain general properties of these concepts are studied. The weak edge detour numbers of certain classes of graphs are determined. Its relationship with the detour diameter is discussed and it is proved that for each triple \(D, k, p\) of integers with \(8 \leq k \leq p-D+1\) and \(D \geq 3\) there is a connected graph \(G\) of order \(p\) with detour diameter \(D\) and \(dn_w(G) = k\). It is also proved that for any three positive integers \(a, b, k\) with \(k \geq 3\) and \(a \leq b \leq 2a\), there is a connected graph \(G\) with detour radius \(a\), detour diameter \(b\) and \(dn_w(G) = k\). Graphs \(G\) with detour diameter \(D \leq 4\) are characterized for \(dn_w(G) = p-1\) and \(dn_w^+(G) = p-2\) and trees with these numbers are characterized. A weak edge detour set \(S\), no proper subset of which is a weak edge detour set, is a minimal weak edge detour set. The upper weak edge detour number \(dn_w^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal weak edge detour set of \(G\). It is shown that for every pair \(a, b\) of integers with \(2 \leq a \leq b\), there is a connected graph \(G\) with \(dn_w(G) = a\) and \(dn_w^+(G) = b\).

The vertex Padmakar-Ivan \((PI_v)\) index of a graph \(G\) is defined as the summation of the sums of \([m_{eu}(e|G) + m_{eu}(e|G)]\) over all edges \(e = uv\) of a connected graph \(G\), where \(m_{eu}(e|G)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\), and \(m_{eu}(e|G)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). In this paper, we give the explicit expressions of the vertex PI indices of some sums of graphs.

A graph \(G\) is called \(H\)-equicoverable if every minimal \(H\)-covering in \(G\) is also a minimum \(H\)-covering in \(G\). In this paper, we give the characterization of connected \(M_2\)-equicoverable graphs with circumference at most \(5\).

In this paper, we investigate the existence of \(2\)-\((v,8,1)\) designs admitting a block-transitive automorphism group \(G \leq \mathrm{ATL}(1,q)\). Using Weil’s theorem on character sums, the following theorem is proved:If a prime power \(q\) is large enough and \(q \equiv 57 \pmod{112}\), then there is always a \(2-(v,8,1)\) design which has a block-transitive, but non flag-transitive automorphism group \(G.\)