Generalised nice sets are defined as subsets of edges of the extended Fano plane satisfying a certain absorbing property. The classification up to collineations, purely combinatorial in nature, provides 245 generalised nice sets. In turn, this gives rise to new Lie algebras obtained by modifying the bracket of homogeneous elements on an initial Z32-graded Lie algebra.

<p>In this paper, we explore some interesting applications of the matrix tree theorem. In particular, we present a combinatorial interpretation of a distribution of <span class="math inline">\((n-1)^{n-1}\)</span>, in the context of uprooted spanning trees of the complete graph <span class="math inline">\(K_{n}\)</span>, which was previously obtained by Chauve–Dulucq–Guibert. Additionally, we establish a combinatorial explanation for the distribution of <span class="math inline">\(m^{n-1}n^{m-1}\)</span>, related to spanning trees of the complete bipartite graph <span class="math inline">\(K_{m,n}\)</span>, which seems new. Furthermore, we extend this study to the graph <span class="math inline">\(K_{n}\setminus \{e_{1,n}\}\)</span>, obtained by deleting an edge from <span class="math inline">\(K_n\)</span>, and derive a new identity for the number of its uprooted spanning trees.</p>

<p>The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic number game is played on a graph <span class="math inline">\(G\)</span> by two players, Alice and Bob, who take turns selecting a vertex and placing it into one of <span class="math inline">\(k\)</span> sets. Alice is trying to make each of these sets into a dominating set of <span class="math inline">\(G\)</span> while Bob’s goal is to prevent this from being accomplished. The maximum <span class="math inline">\(k\)</span> for which Alice can achieve her goal when both players are playing optimal strategies, is called the game domatic number of <span class="math inline">\(G\)</span>. There are two versions of the game and two resulting invariants depending on whether Alice or Bob is the first to play. We prove several upper bounds on these game domatic numbers of arbitrary graphs and find the exact values for several classes of graphs including trees, complete bipartite graphs, cycles and some narrow grid graphs. We pose several open problems concerning the effect of standard graph operations on the game domatic number as well as a vexing question related to the monotonicity of the number of sets available to Alice.</p>

This paper investigates the number of rooted biloopless nonseparable planar near-triangulations and presents some formulae for such maps with three parameters: the valency of root-face, the number of edges and the number of inner faces. All of them are almost summation-free.

In this paper, we give a classification of all Mengerian 4-uniform hypergraphs derived from graphs.

Let be a simple connected graph with vertex set and edge set . The Randić index of graph is the value , where and refer to the degree of the vertices and . We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.

In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected ( H C ) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.

This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of V and W . These values are crucial in determining the Hamming distance and MDS codes in the family of certain constacyclic codes over F p m [ u ] / ⟨ u 3 ⟩ , which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.

For a graph G and for non-negative integers p , q and r , the triplet ( p , q , r ) is said to be an admissible triplet, if 3 p + 4 q + 6 r = | E ( G ) | . If G admits a decomposition into p cycles of length 3 , q cycles of length 4 , and r cycles of length 6 for every admissible triplet ( p , q , r ) , then we say that G has a { C p 3 , C q 4 , C r 6 } -decomposition. In this paper, the necessary conditions for the existence of { C p 3 , C q 4 , C r 6 } -decomposition of K ℓ , m , n ( ℓ ≤ m ≤ n ) are proved to be sufficient. This affirmatively answers the problem raised in \emph{Decomposing complete tripartite graphs into cycles of lengths 3 and 4 , Discrete Math. 197/198 (1999), 123-135}. As a corollary, we deduce the main results of \emph{Decomposing complete tripartite graphs into cycles of lengths 3 and 4 , Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths 3 and 6 , Austral. J. Combin., 73(1), 220-241 (2019)}.

A chemical structure specifies the molecular geometry of a given molecule or solid in the form of atom arrangements. One way to analyze its properties is to simulate its formation as a product of two or more simpler graphs. In this article, we take this idea to find upper and lower bounds for the generalized Randić index \(\mathcal{R}_{\alpha}\) of four types of graph products, using combinatorial inequalities. We finish this paper by providing the bounds for \(\mathcal{R}_{\alpha}\) of a line graph and rooted product of graphs.