Let \(G\) be a graph of order \(n\) and let \(\mu\) be an eigenvalue of multiplicity \(m\). A star complement for \(\mu\) in \(G\) is an induced subgraph of \(G\) of order \(n-m\) with no eigenvalue \(\mu\). Some general observations concerning graphs with the complete tripartite graph \(K_{r,s,t}\) as a star complement are made. We study the maximal regular graphs which have \(K_{r,s,t}\) as a star complement for eigenvalue \(\mu\). The results include a complete analysis of the regular graphs which have \(K_{n,n,n}\) as a star complement for \(\mu = 1\). It turns out that some well-known strongly regular graphs are uniquely determined by such a star complement.

In this paper, we first prove that if the edges of \(K_{2m}\) are properly colored by \(2m-1\) colors in such a way that any two colors induce a 2-factor of which each component is a 4-cycle, then \(K_{2m}\) can be decomposed into \(m\) isomorphic multicolored spanning trees. Consequently, we show that there exist three disjoint isomorphic multicolored spanning trees in any properly \((2m-1)\)-edge-colored \(K_{2m-1}\) for \(m \geq 14\).

The Merrifield-Simmons index, denoted by \(i(G)\), of a graph \(G\) is defined as the total number of its independent sets. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than \(3\) in its unique cycle. Let \(\mathcal{U}_n^1\) be the set of fully loaded unicyclic graphs. In this paper, we determine graphs with the largest, second-largest, and third-largest Merrifield-Simmons index in \(\mathcal{U}_n^1\).

For a graph \(G = (V, E)\), the modified Schultz index of \(G\) is defined as \(S^0(G) = \sum\limits_{\{u,v\} \subset V(G)} (d_G(u) – d_G(v)) d_{G}(u, v)\), where \(d_G(u)\) (or \(d(u)\))is the degree of the vertex \(u\) in \(G\), and \(d_{G}(u, v)\) is the distance between \(u\) and \(v\). The first Zagreb index \(M_1\) is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index \(M_2\) is equal to the sum of the products of the degrees of pairs of adjacent vertices. In this paper, we present a unified approach to investigate the modified Schultz index and Zagreb indices of tricyclic graphs. The tricyclic graph with \(n\) vertices having minimum modified Schultz index and maximum Zagreb indices are determined.

Let \(T = (V, A)\) be a (finite) tournament and \(k\) be a non-negative integer. For every subset \(X\) of \(V\)\), the subtournament \(T[X] = (X, A \cap (X \times X))\) of \(T\), induced by \(X\), is associated. The dual tournament of \(T\), denoted by \(T^*\), is the tournament obtained from \(T\) by reversing all its arcs. The tournament \(T\) is self-dual if it is isomorphic to its dual. \(T\) is \((-k)\)-self-dual if for each set \(X\) of \(k\) vertices, \(T[V \setminus X]\) is self-dual. \(T\) is strongly self-dual if each of its induced subtournaments is self-dual. A subset \(I\) of \(V\) is an interval of \(T\) if for \(a,b \in I\) and for \(x \in V \setminus I\), \((a,x) \in A\) if and only if \((b,x) \in A\). For instance, \(\emptyset\), \(V\), and \(\{x\}\), where \(x \in V\), are intervals of \(T\) called trivial intervals. \(T\) is indecomposable if all its intervals are trivial; otherwise, it is decomposable. A tournament \(T’\), on the set \(V\), is \((-k)\)-hypomorphic to \(T\) if for each set \(X\) on \(k\) vertices, \(T[V \setminus X]\) and \(T'[V \setminus X]\) are isomorphic. The tournament \(T\) is \((-k)\)-reconstructible if each tournament \((-k)\)-hypomorphic to \(T\) is isomorphic to it.

Suppose that \(T\) is decomposable and \(|V| \geq 9\). In this paper, we begin by proving the equivalence between the \((-3)\)-self-duality and the strong self-duality of \(T\). Then we characterize each tournament \((-3)\)-hypomorphic to \(T\). As a consequence of this characterization, we prove that if there is no interval \(X\) of \(T\) such that \(T[X]\) is indecomposable and \(|V \setminus X| \leq 2\), then \(T\) is \((-3)\)-reconstructible. Finally, we conclude by reducing the \((-3)\)-reconstruction problem.

For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph. In this paper, we characterize the potentially \(C_{2,6}\)-graphic sequences. This characterization partially answers Problem 6 in Lai and Hu [12].

We investigate two modifications of the well-known irregularity strength of graphs, namely the total edge irregularity strength and the total vertex irregularity strength.
In this paper, we determine the exact value of the total edge (vertex) irregularity strength for Halin graphs.

A signed \(k\)-dominating function of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \{+1,-1\}\) such that \(\sum_{u \in N_G[v]} f(u) \geq k\) for each vertex \(v \in V\). A signed \(k\)-dominating function \(f\) of a graph \(G\) is minimal if no \(g \leq f\) is also a signed \(k\)-dominating function. The weight of a signed \(k\)-dominating function is \(w(f) = \sum_{v \in V} f(v)\). The upper signed \(k\)-domination number \(\Gamma_{s,k}(G)\) of \(G\) is the maximum weight of a minimal signed \(k\)-dominating function on \(G\). In this paper, we establish a sharp upper bound on \(\Gamma _{s,k}(G)\) for a general graph in terms of its minimum and maximum degree and order, and construct a class of extremal graphs which achieve the upper bound. As immediate consequences of our result, we present sharp upper bounds on \(\Gamma _{s,k}(G)\) for regular graphs and nearly regular graphs.

The paper contains enumerative combinatorics for positive braids, square free braids, and simple braids, emphasizing connections with classical Fibonacci sequence.