For $i\geq 0$, the $i$-cube $Q_i$ is the graph on $2^i$ vertices representing $0/1$ tuples of len... more For $i\geq 0$, the $i$-cube $Q_i$ is the graph on $2^i$ vertices representing $0/1$ tuples of length $i$, where two vertices are adjacent whenever the tuples differ in exactly one position. (In particular, $Q_0 = K_1$.) Let $\alpha_i(G)$ be the number of induced $i$-cubes of a graph $G$. Then the cube polynomial $c(G,x)$ of $G$ is introduced as $\sum_{i\geq 0} \alpha_i(G) x^i$. It is shown that any function $f$ with two related, natural properties, is up to the factor $f(Q_0,x)$ the cube polynomial. The derivation $\partial\, G$ of a median graph $G$ is introduced and it is proved that the cube polynomial is the only function $f$ with the property $f'(G,x)= f(\partial\, G, x)$ provided that $f(G,0)=|V(G)|$. As the main application of the new concept, several relations that widely generalize previous such results for median graphs are proved. For instance, it is shown that for any $s\geq 0$ we have $c^{(s)}(G,x+1) = \sum_{i\geq s}\, {{c^{(i)}(G,x)}\over {(i-s)!}}\,,$ where certai...
A rich structure theory has been developed in the last three decades for graphs embeddable into h... more A rich structure theory has been developed in the last three decades for graphs embeddable into hypercubes, in particular for isometric subgraphs of hypercubes, also known as partial cubes, and for median graphs. Median graphs, which constitute a proper ...
Balaban index is defined as J ( G ) = m m − n + 2 Σ 1 w ( u ) ⋅ w ( v ) , $J\left( G \right)=\fra... more Balaban index is defined as J ( G ) = m m − n + 2 Σ 1 w ( u ) ⋅ w ( v ) , $J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each n, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.
For $i\geq 0$, the $i$-cube $Q_i$ is the graph on $2^i$ vertices representing $0/1$ tuples of len... more For $i\geq 0$, the $i$-cube $Q_i$ is the graph on $2^i$ vertices representing $0/1$ tuples of length $i$, where two vertices are adjacent whenever the tuples differ in exactly one position. (In particular, $Q_0 = K_1$.) Let $\alpha_i(G)$ be the number of induced $i$-cubes of a graph $G$. Then the cube polynomial $c(G,x)$ of $G$ is introduced as $\sum_{i\geq 0} \alpha_i(G) x^i$. It is shown that any function $f$ with two related, natural properties, is up to the factor $f(Q_0,x)$ the cube polynomial. The derivation $\partial\, G$ of a median graph $G$ is introduced and it is proved that the cube polynomial is the only function $f$ with the property $f'(G,x)= f(\partial\, G, x)$ provided that $f(G,0)=|V(G)|$. As the main application of the new concept, several relations that widely generalize previous such results for median graphs are proved. For instance, it is shown that for any $s\geq 0$ we have $c^{(s)}(G,x+1) = \sum_{i\geq s}\, {{c^{(i)}(G,x)}\over {(i-s)!}}\,,$ where certai...
A rich structure theory has been developed in the last three decades for graphs embeddable into h... more A rich structure theory has been developed in the last three decades for graphs embeddable into hypercubes, in particular for isometric subgraphs of hypercubes, also known as partial cubes, and for median graphs. Median graphs, which constitute a proper ...
Balaban index is defined as J ( G ) = m m − n + 2 Σ 1 w ( u ) ⋅ w ( v ) , $J\left( G \right)=\fra... more Balaban index is defined as J ( G ) = m m − n + 2 Σ 1 w ( u ) ⋅ w ( v ) , $J\left( G \right)=\frac{m}{m-n+2}\Sigma \frac{1}{\sqrt{w\left( u \right)\cdot w\left( v \right)}},$ where the sum is taken over all edges of a connected graph G, n and m are the cardinalities of the vertex and the edge set of G, respectively, and w(u) (resp. w(v)) denotes the sum of distances from u (resp. v) to all the other vertices of G. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each n, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.
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Papers by Riste Škrekovski