The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum
of weights \(\frac{2}{d(u) + d(v)}\) of all edges \(uv\) of \(G\), where
\(d(u)\) denotes the degree of a vertex \(u\) in \(G\).

In this paper, we establish sharp lower and upper bounds for the
harmonic index of bicyclic graphs and characterize the
corresponding extremal graphs.

For a graph \(G\), its Hosoya index is defined as the total number
of matchings in it, including the empty set. As one of the oldest and
well-studied molecular topological descriptors, the Hosoya index has
been extensively explored.

Notably, existing literature has primarily focused on its extremal
properties. In this note, we bridge a significant gap by establishing
sharp lower bounds for the Hosoya index in terms of other topological
indices.

We present a unified extension of alternating subsets to \(k\)-combinations
of \(\{1, 2, \ldots, n\}\) containing a prescribed number of sequences
of elements of the same parity. This is achieved by shifting attention
from parity-alternating elements to pairs of adjacent elements of the
same parity.

Enumeration formulas for both linear and circular combinations are
obtained by direct combinatorial arguments. The results are applied
to the enumeration of bit strings.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\).
Define the orientation number of \(G\), \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\),
where \(d(D)\) denotes the diameter of the digraph \(D\).

In this paper, it is shown that \(\overrightarrow{d}(G(n_1, n_2, \ldots, n_p)) = d(G)\),
where \(G(n_1, n_2, \ldots, n_p)\) is a \(G\)-vertex multiplication
([2]) of a connected bipartite graph \(G\) of order \(p \geq 3\)
with diameter \(d(G) \geq 5\) and any finite sequence \(\{n_1, n_2, \ldots, n_p\}\)
with \(n_i \geq 3\).

Cyclic frames, or partially partition-type cyclic relative difference
families, are combinatorial structures that are used to produce series
of optimal families consisting of a single frequency hopping sequence
and optimal difference systems of sets for code synchronization.

In this paper, two new classes of cyclic frames from finite geometries
are obtained.

Consider the game of locating a marked vertex on a connected graph,
where the player repeatedly chooses a vertex of the graph as a probe,
and is given the distance from the probe to the marked vertex,
until she can uniquely locate the hidden vertex. The goal is to
minimize the number of probes.

The static version of this game is the well-known problem of finding
the metric dimension (or location number ) of the graph.
We study the sequential version of this game, and the corresponding
sequential location number .

We establish several formulae for sums and alternating sums of products
of generalized Fibonacci and Lucas numbers. In particular, we extend
some results of Z. Cerin and of Z. Cerin and G. M. Gianella .

An \({H}_2\) graph is a multigraph on three vertices with a double
edge between a pair of distinct vertices and single edges between
the other two pairs. In this paper, we settle the \({H}_2\) graph
decomposition problem, which was left unfinished in a paper of
Hurd and Sarvate, by decomposing a complete multigraph \(3K_{8t}\)
into \({H}_2\) graphs recursively.

This article is a contribution to the study of the automorphism groups
of \(2\)-\((v,k,1)\) designs. Let \(\mathcal{D}\) be a \(2\)-\((v,13,1)\) design and
suppose that \(G\) is a group of automorphisms of \(\mathcal{D}\) which is
block-transitive and point-primitive. Then \(\mathrm{Soc}(G)\),
the socle of \(G\), is not isomorphic to \(^2G_2(q)\) or to \(^2F_4(q^2)\)
for any prime power \(q\).

Let \(G\) be a finite permutation group acting primitively on sets \(\Omega_1\) and \(\Omega_2\). We describe a construction of a \(1\)-design
with the block set \(\mathcal{B}\) and the point set \(\Omega_2\), having \(G\) as an automorphism group.Applying this method, we construct a unital \(2\)-\((q^3+1, q+1, 1)\) design and a semi-symmetric design \((q^4-q^3+q^2, q^2-q, (1))\) from the unitary group \(U(3,q)\), where \(q = 3, 4, 5, 7\).From the unital and the semi-symmetric design, we build a projective plane \(PG(2,q^2)\). Further, we describe other combinatorial structures constructed from these unitary groups.