A star edge coloring of a graph \(G\) is a proper edge coloring in which every path or cycle on four vertices contains at least three distinct colors. The star edge chromatic number of \(G\), denoted by \(\chi^\prime_{\mathrm{st}}(G)\), is the minimum number of colors required for such a coloring. In this paper, we study the star edge chromatic number of several graph classes derived from the Knödel graph \(W_{\Delta,n}\). In particular, we determine bounds and exact values for \(\chi^\prime_{\mathrm{st}}(G)\) for the Knödel graph \(W_{\Delta,n}\), its middle graph \(M\!\left(W_{\Delta,n}\right)\), and its Mycielskian \(\mu\!\left(W_{\Delta,n}\right)\). Furthermore, we provide explicit constructions of star edge colorings that attain these values.

Let \(G\) be a simple connected mixed graph, and let \(H(G)\) denote the Hermitian adjacency matrix of \(G\). The Hermitian permanental polynomial of \(G\) is defined as \(\pi(G; x) = \operatorname{per}(xI – H(G))\), where \(\operatorname{per}(\cdot)\) represents the permanent and \(I\) is the identity matrix. In this paper, we first derive fundamental properties of the Hermitian permanental polynomial for mixed graphs and establish explicit formulas relating its coefficients to those of the characteristic polynomial. We then analyze the root distribution of this polynomial, determining the number of zero roots for several special classes of mixed graphs. Finally, we characterize mixed graphs that remain cospectral under four‑way switching and prove that this operation preserves the permanental spectrum.

The Euler Sombor (\(EU\)) index of a graph \(G\) is defined as \[EU(\mathit{G})=\sum \limits_{{\mathit{u}}{\mathit{v}}\in E(\mathit{G})} \sqrt{deg_G^2(u)+deg_G^2(v)+deg_G(u)deg_G(v)},\] where \(deg_G(u)\) and \(deg_G(v)\) are the degrees of the vertices \(u\) and \(v\) in the graph \(G\), respectively. Biswaranja Khanra and Shibsankar Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun. Math. Comput. Chem., 94 (2025) 525–548] posed an open problem to determine the extremal values and extremal graphs of the Euler Sombor index in the class of all connected graphs with a given domination number. In this paper, we solve this open problem for trees with a given domination number. Furthermore, we determine an upper bound for the Euler Sombor index of trees with a given independence number. We also characterize the corresponding extremal trees. Additionally, we propose a set of open problems for future research.

The exponential Randić index of a graph \(G\), denoted by \(ER(G)\), is defined as \(\sum\limits_{uv\in E(G)}e^{\frac{1}{d(u)d(v)}}\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). The line graph \(L(G)\) of a graph \(G\) is a graph in which each vertex represents an edge of \(G\), and two vertices are adjacent in \(L(G)\) if and only if their corresponding edges in \(G\) are incident to a common vertex. In this paper, we proved that for any tree \(T\) of order \(n\ge3\), \(ER(L(T))>\frac{n}{4}e^{\frac{1}{2}}\).

We prove the following necessary conditions for signed graphs on complete graphs to admit additive graceful labelings, thus considerably narrowing down the search for such labelings. If a signed graph on a complete graph \(K_p, ~p\not = 4\), with \(n\) negative and \(m\) positive edges, admits an additively graceful labeling, then (1) \(p\), \(p-2\) or \(p-4\) must be a perfect square, (2) If \(p\) is a perfect square then, both \(m\) and \(n\) are even, (3) If \(p-4\) is a perfect square then, both \(m\) and \(n\) are odd, (4) If \(p-2\) is a perfect square then, \(m\) and \(n\) have opposite parity and (5) The number of odd and even labeled vertices in \(S\) are \(\frac{p+\sqrt{p-i}}{2}\) and \(\frac{p-\sqrt{p-i}}{2}\) according as \(p-i\) is a perfect square, \(i=0,2\) or \(4\). We also provide an example to show that, if a signed graph \(S\) with no negative edges is additively graceful then, \(S\) as a graph need not be graceful. Interestingly, this example also settles 3 more open problems concerning gracefulness and edge consecutive gracefulness (ecg), thereby demonstrating that ecg or \(m\)-gracefulness is not a reliable measure of gracefulness.

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact enumeration of (3, 6)-fullerenes with V vertices. We also enumerate (3, 6)-fullerenes with mirror symmetry, with 3-fold rotational symmetry, and with both types of symmetry. The resulting formulas are expressed in terms of the prime factorization of V.

Given any integers \(q\ge 2\) and \(p\ge 3\), a multidimensional array \[[m(i_1,i_2, \dots, i_q)\colon 1\le i_1\le p, \dots , 1\le i_q\le p],\] with non-repeated entries from the set \(\{1,2,\dots, p^q\}\) will be called a \(q\) dimensional magic hyper-square of order \(p\) if the sum of the numbers in any of its axes or diagonals is \(p(p^q+1)/2\). In this paper, we study odd-order uniform step magic hyper-squares of the form \[m(i_1, \dots, i_q)=1+\sum\limits_{j=1}^{q} p^{j-1}\left[\left(\sum\limits_{k=1}^qa_{j,k} i_k+d_j\right) \bmod p \right] .\] We derive necessary and sufficient conditions for these coefficients to generate magic hyper-squares and use a specific choice of coefficients to get new formula that generates magic hyper-squares of all odd orders greater than two.

In this paper, we consider two ways of breaking a graph’s symmetry: distinguishing labelings and fixing sets. A distinguishing labeling ϕ of G colors the vertices of G so that the only automorphism of the labeled graph (G, ϕ) is the identity map. The distinguishing number of G, D(G), is the fewest number of colors needed to create a distinguishing labeling of G. A subset S of vertices is a fixing set of G if the only automorphism of G that fixes every element in S is the identity map. The fixing number of G, Fix(G), is the size of a smallest fixing set. A fixing set S of G can be translated into a distinguishing labeling ϕ_S by assigning distinct colors to the vertices in S and assigning another color (e.g., the “null” color) to the vertices not in S. Color refinement is a well-known efficient heuristic for graph isomorphism. A graph G is amenable if, for any graph H, color refinement correctly determines whether G and H are isomorphic or not. Using the characterization of amenable graphs by Arvind et al. as a starting point, we show that both D(G) and Fix(G) can be computed in O((|V(G)|+|E(G)|)log |V(G)|) time when G is an amenable graph.

This paper studies the classification problem of block-transitive \( t \)-designs. Let \(\cal D = (\mathcal{P}, \mathcal{B}) \) be a non-trival \( t\)-\((v,k,\lambda) \) design with \( \lambda \leq 5 \), and let \( G \) be a block-transitive, point-primitive automorphism group of \(\cal D \). We prove that if \( \text{Soc}(G) \) is a sporadic simple group, then up to isomorphism, there are exactly 15 such designs \( \cal D \).

The Mostar invariants are newly introduced bond-additive, distance-related descriptors that compute the degree of peripherality of specific edges as well as the entire graph. These invariants have attracted significant attention in both classical applications of chemical graph theory and studies of complex networks. They have proven to be useful for exploring the topological aspects of these networks. For a graph ℋ, the edge Mostar index Mo_e is defined as the sum of the magnitudes of the differences between m_ℋ(x) and m_ℋ(g) across all edges xg of ℋ. Here, m_ℋ(g) (or m_ℋ(x)) represents the cardinality of the edges in ℋ that are closer to g (or x) than x (or g). In this paper, we determine the trees that maximize and minimize the edge Mostar index for fixed order, diameter, and number of pendent vertices. Sharp upper and lower bounds for this index are established, and the corresponding extremal trees are characterized. Moreover, the correlation of the edge Mostar index with certain physicochemical properties is examined.